Notes of Will Sawin first Hadamard lecture, 15-05-2023

Number theory over function fields

1. The classical theory

Let us start with classical stuff.

Theorem 1 (Dirichlet) a positive integer, an integer. There exist infinitely many prime numbers such that if and only if .

A more precise statement is the prime number theorem on arithmetic progressions:

for some explicit .

In other words, remainders mod of primes are evenly distributed.

The proofs of both theorems rely on properties of Dirichlet characters.

Definition 2 A function is a Dirichlet character mod if

1. for all ,
2. for all ,
3. .

From a Dirichlet character, one constructs a Dirichlet -function

1.1. Properties

If , the trivial character, then is entire. Dirichlet’s theorem follows from . The prime number theorem on arithmetic progressions follows from for all and on a neighborhood of that line. Improving the remainder to be polynomial of degree is amounts to nonvanishing of on . This is hard. The special case is known as Generalized Riemann Hypothesis.

If one thinks of primes as random, i.e. an integer has a probability to be prime which is , the expected error from this random model is .

Other interesting questions about Dirichlet -functions are statistical: what happens on the average over ?

1.2. Example: moments

We are interested in

An exact expression is known only for , or , .

2. Function fields

These problems being too hard, I will study similar questions in a different, hopefully easier, setting. Let denote a field with elements, denotes the ring of polynomials with coefficients in . I intend to replace integers with . For instance, the Euclidean algorithm works on polynomials.

I denote by the set of monic polynomials (thought of as an analogue of positive integers).

For , its absolute value is , which is equal to the cardinality of the quotient ring .

The advantage of the this shift of setting is that new connections with other fields of mathematics appear. Today, I will give one instance of that.

2.1. Dirichlet characters

Say that a function is a Dirichlet character mod if

1. for all ,
2. for all ,
3. .

The Dirichlet -function is

It is a power series in . is entire for , because it is a polynomial in . Indeed,

and the sum vanishes for , since each residue class mod occurs times and, by orthogonality of characters, .

The Euler product formula holds,

showing that for . If we can improve nonvanishing we get information on the number of irreducible such that and .

2.2. Generalized Riemann Hypothesis

In the new setting, the analogue of the Generalized Riemann Hypothesis is known, this is

Theorem 3 (Weil) for .

The proof is geometric.

Weil’s theorem implies that

2.3. Statistical issues

The theorem leaves open statistical questions about . Katz, Katz-Sarnak, Deligne answered statistical questions in the limit where tends to infinity.

Definition 4 Say a character is primitive if there is no character of smaller modulus such that for all with .

Say is odd if for some .

Theorem 5 Assume is primitive and odd. Then is a polynomial in of degree exactly with all roots on . It follows that is the characteristic polynomial of where is a unitary matrix.

Theorem 6 (Katz) If is squarefree and , there a exists a map

which is continuous, conjugacy invariant, such that

In other words, the equidistribute in the unitary group as tends to infinity. It kills hope to control its nonvanishing simultaneously for all .

2.4. A sample result

Here is a theorem of mine.

Theorem 7 Let be squarefree of degree . Then

In other words, the average over primitive, odd characters is 1 up to fluctuations which become small if .

Whereas, in the classical setting, only a few moments are understood, in the new setting one can study arbitrarily high moments, provided is large enough.

The question has a geometric nature because it involves counting solutions to polynomial equations over . Since the middle of XXth century, one knows that such a counting can follow from the same topological techniques used to describe solutions over the complex numbers.

2.5. Scheme of proof

Let me give a scheme of the proof of Theorem 7. By the Euler product formula, the lefthand sum can be rewritten

For a fixed ,

Therefore the sum becomes a sum over involving the number of such that and .

This set is the union over -tuples of natural numbers summing to of sets, each of which is the set of solutions to equations in variables, over (the unknowns are the coefficients of the monic polynomials ).

The answer, obtained by Grothendieck’s school, can be found on the windows of Orsay’s math building.

Theorem 8 (Grothendieck-Lefschetz formula) Let be a scheme of finite type over . Then the number of points of over is

We also need

Theorem 9 (Deligne’s Riemann hypothesis) The eigenvalues of on are algebraic integers of absolue value .

This implies that the trace is bounded above by times the dimension of cohomology. This allows to conclude

Theorem 10 The number of solutions to the counting problem is

Thus we see that geometry has made it possible to go beyond the classical results in the function field setting.

Here is the key geometric statement. Let be a monic squarefree polynomial of degree over a field . Let be prime to . Consider the union of the schemes parametrizing the set of whose product equals . Then the cohomology of the points over splits as the sum of a boring piece (independent on ) and an interesting piece which is -dimensional if , and otherwise its dimension is at most

Notes on Alexandros Eskenazis Rennes lectures january 26th 2023

When does embed into ?

The question goes back to Banach. This will be an excuse to review some modern tools. Banach space theory has evolved into metric geometry. The leitmotiv will be: how does one prove non-embeddability?

1. Linear embeddings

1.1. Banach’s question

Notation. measurable functions on . -summable sequences.

in its norm.

Definition 1 A linear operator between normed spaces is a -isomorphic embedding, where , if there exists such that ,

Example 1 embeds in isometrically.

Remark. is not isomorphic to when . This is a nontrivial fact, probably going back to Banach.

Question (Banach). For which does embed isomorphically into ?

Theorem 2 (Banach 1932, Paley 1936) does not embed into unless or .

Theorem 3 (Kadec 1958) embeds isometrically into if or .

Theorem 2 is much harder than Theorem 3 (although older).

1.2. Proof of Kadec’ theorem

Let be a sequence of iid standard gaussian random variables on a -finite probability space.

Gaussians have the following property: if standard gaussian r.v. and are independent, the random variable has the same distribution as . Indeed, both have the same characteristic function.

Define the embedding as follows,

Then has the same distribution as . Therefore

Question. Why doesn’t this work for other values of ? In other words, do there exist iid symmetric random variables such that has the same distribution as ?

The answer arises in Paul Lévy’s book.

Theorem 4 (Levy 1951) Such random variables exist iff . They are called standard -stable random variables, and for , they satisfy

Therefore,

is finite iff .

This shows that embeds isometrically into for .

Remark 1 To prove that embeds into in this range, one needs two more nontrivial results:

• is finitely representable into .
• If is finitely representable in and is separable, then embeds into .

This ends our excursion in the construction of embeddings. From now on, we switch to nonembeddability results.

1.3. Linear distorsion

Definition 5 The linear distorsion of into is the smallest such that there exists a -isomorphic embedding of into . It is denoted by

and if , by

Thus we have stated and partly proved that

Goal. Understand the asymptotics of when are in the second range.

Theorem 6 Let . Then

Easy. The upper bounds are easy, the embedding is identity or identity to followed with Kadec’s isometric embedding.

Our goal is to prove lower bounds, by designing invariants.

2. Smoothness and convexity in spaces

2.1. Smoothness and convexity

Definition 7 (Ball – Carlen – Lieb 1993) Fix . Say a Banach space is -uniformly smooth with constant if

Fix . Say a Banach space is -uniformly convex with constant if

The best constants are denoted by and .

Exercise. (Lindenstrauss’ duality formula à la Ball-Carlen-Lieb). For any normed space and ,

2.2. The case of spaces

Theorem 8 (Clarkson’s inequality) For , .

For , .

Proof. Since the inequality involves only , is suffices to prove that for all ,

By monotonicity of norms and the parallelogram identity,

Theorem 9 For , .

For , .

2.3. Two lemmata

The proof of Theorem 10 requires two lemmata, the first one is Bonami’s hypercontractivity inequality on the -cube.

Lemma 10 (Bonami’s two-point inequality) Let . For ,

Proof. One can assume that and . Using Taylor’s expansion, one checks that

The second lemma illustrates a general principle: an inequality for with constant one can be nothing but a relaxation of the parallelogram identity in Euclidean space.

Lemma 11 (Hanner’s inequality) Let . For ,

One first checks that for , the numbers

satisfy

(again, one can assume that and ; then it amounts to the monotonicity of a function).

Hanner’s inequality follows: assuming that , set , , to get

Integrating with respect to yields

2.4. Proof of Theorem 10

Let , . We show that

Indeed, first apply Bonami’s inequality.

Conjecture. Does Hanner’s inequality hold in Schatten class, i.e. for matrices where the norm of singular values is used?

Known for and (Ball-Carlen-Lieb). Heinavaara 2022 proves this for .

Conjecture. For ,

Theorem 12 (D. Schechtman 1995) Yes for .

3. Martingales in Banach spaces

Definition 13 Let be a Banach space. A (Paley-Walsh) martingale with values in is a sequence of functions such that for all ,

The basic example is

for given vectors .

Remark. If , the set is orthogonal, so

Definition 14 (Pisier 1975) A Banach space has martingale type , , il for every -valued martingale ,

has martingale cotype , , il for every -valued martingale ,

These properties can be thought of as relaxations of the identity that holds in Hilbert space.

3.1. Smoothness/convexity versus type/cotype

These properties follow from the convexity properties introduced earlier.

Proposition 15 (Pisier) -smoothness (resp. -convexity) implies martingale type (resp. cotype ) with the same constant.

Pisier’s renorming theorem states that the converse is true, up to changing for an equivalent norm.

The advantage of the type/cotype formulation is that the error is multiplicative, hence these properties are isomorphism invariant.

3.2. Proof of Pisier’s “smoothness implies type”

Let be an -valued martingale with . Then

thus has martingale type with constant .

Corollary 16 Let .

If , then

If , then

Beware that has Rademacher cotype with constant but no nontrivial martingale cotype.

3.3. Proof of Theorem 7, type and cotype range

Let be an embedding of distorsion . Apply type of to . Then

On the other hand,

This yields .

The proofs of the three other lower bounds on are similar. The results are sharp.

Note that the argument used linearity very strongly.

4. Nonlinear embeddings

Given metric spaces and , one can speak of the biLipschitz distorsion of into , denoted by , as the least such that there exists and satisfying

Now we discretize spaces. Let , viewed as an approximation of . What can one say about ?

Theorem 17 Let . Then is of the order of

So we see that a phase transition occurs when .

In certain cases, the upper bounds are smart, but I will not focus on them.

In the unknown range , the best we know now is

The left-hand side would be sharp if the following was true: for every and every , the metric space has a biLipschitz embedding into .

Theorem 18 (Bretagnolle – Dacunha-Castelle – Krivine 1965) This is true for .

In my thesis, we have the following result:

Theorem 19 (Eskenazis – Naor 2016) For , , , the space

does not embed in .

5. Metric type

5.1. Enflo type

Definition 20 (Enflo 1969, modern terminology) A metric space has Enflo type with constant if for all , for all ,

Remark. If is a normed space and , then the lefthand side is

and the righthand side is

so this is really a metric analogue of Rademacher type: for linear spaces,

Theorem 21 (Khot – Naor 2006) For normed linear spaces,

Proof. Let which has martingale type . Define a martingale

From martingale type, we know that . But

where has one sign changed. So

where has length and one sign flipped at position . So summing over yields the expected righthand side. Now add and subtract the expectation,

which completes the proof.

The converse is a recent breakthrough, still in the linear case:

Theorem 22 (Ivanisvili – von Handel – Volberg 2020) For linear spaces, Rademacher type implies Enflo type.

5.2. Proof of distorsion lower bounds for

The upper bound is easy (use global embedding of ).

Here is another easy lower bound:

Assume that (the other case is similar). Let have distorsion with rescaling . Since has Enflo type with constant ,

(one sign change in ), so the righthand side is bounded above by , and the lefthand side is bounded below by , so .

6. Metric cotype

Formally, Rademacher cotype is the reverse of Rademacher type, but a naive delinearization does not work. The cube does not suffice, one needs an extra scaling factor .

Definition 23 (Mendel – Naor 2008) A metric space has metric cotype with constant if for any , there exists such that any function satisfies

Remark. As soon as has at least 2 points, .

Theorem 24 (Mendel – Naor 2008, Giladi – Mendel – Naor 2011) For normed spaces, Rademacher cotype is equivalent to metric cotype, and one can always take .

The major open question is wether one can take . Indeed, this would have many geometric applications.

Theorem 25 (Eskenazis – Mendel – Naor 2019) For normed spaces, martingale cotype implies metric cotype, with .

6.1. Proof of the remaining distorsion lower bounds in Theorem 18

I will cheat and identify with . This is not a serious matter, since

We must show that

Since decreasing decreases the -distorsion, one can assume that . Let have distorsion . Then the lefthand side of metric cotype assumption is , whereas the right hand side is , so .

We see that we really need .

6.2. Proof of Theorem 26

The strategy is inspired from hypercontractivity: smoothing the space using an averaging operator puts functions closer to linear functions, i.e. leads us closer to the linear setting.

Given , we define a martingale

where has length . We view as a smoothing operator.

If and , let . Then

where (1) is an approximation term,

and (2) is a smoothing term,

We first estimate (1) with Hölder,

Jensen’s inequality gives

In other words, contracts , so

which is ok since , this what we want to see on the righthand side.

Next we bound (2). We can replace with mod . For fixed , write the summand in (2) as a telescopic sum,

But

By martingale cotype,

Adding gives the expected bound .

6.3. Final comment

What does for you is that it provides invariance under coarse embeddings, and not merely biLipschitz embeddings. So the story is far from being finished.

Notes of Maryna Viazovska’s Orsay lecture, january 2023

Sphere packings

This lecture has been designed for an audience of 150 master’s students in Orsay, Universite Paris-Saclay.

1. The sphere packing problem

1.1. Density

In Euclidean space, one tries to pack equal balls as densely as possible. Pack means that interiors of balls are disjoint.

The density of a sphere packing in a box is the ratio

The density of the packing is defined as a , over larger and larger boxes, of densities of packings in boxes.

The supremum of densities of ball packings in is denoted by , this is the sphere packing constant in .

Problem. Determine .

This is a hard problem. It is an instance of a family of similar problems, in other metric spaces.

1.2. Codes

Let be a metric space, and . An -code in is a subset of such that all pairwise distances are .

Example. The centers of a packing of by balls of radius is an -code.

Example. In a graph, equipped with the graph metric (which is integer valued), an independent set is a -code.

1.3. What are -codes good for

Example. Let be the Hamming cube, i.e. and the distance between two points (strings) is the number of coordinates where they differ.

Imagine being given the task of transmitting strings of s and s of length to your friend. You are provided with a device which converts such strings into strings of length , which belong to an -code of . Your friend owns a device that converts back each element of into the -string it arose from. Now you send him converted strings by radio. As soon as the radio transmission introduces no more than errors, your friend can unambiguously recover the -string you broadcast from the noisy string: it is the only element of that sits at distance from it. Thus he can correct the errors. Using his own converter, he can recover the original -string.

Thus -codes in the Hamming cube of high density are desirable, since they allow to transmit long strings without error (after correction).

2. Results in low dimensions

has been guessed for centuries: it is achieved by the hexagonal packing in the plane. The first published proof is due to the norwegian mathematician Axel Thue in 1910, but considered as insufficiently rigorous. In 1943, the hungarian mathematician Laszlo Fejes Toth published a complete proof. A good reference for this proof is Brass, Moser and Pach’s book on Research problems in discrete geometry. Other, simpler, proofs have been found since.

Tom Hales’s determination of requires a lot of 3-dimensional geometry, plus help from a computer.

3. Convex programming

Today, I explain the method of convex programming and how it is used for the packing problem.

How can one replace the problem, which is not convex, by a convex one? By duality, points will be replaced with functions.

3.1. Copositive and positive definite functions

Let be a metric space. Let denote the set of values of .

Definition 1 A function is copositive if

for all finite subsets of .

Copositive function form a convex cone. If we can find a good description of it, we would be in good position to solve the packing problem. Unfortunately, this is rarely the case. Therefore we switch to an easier class of functions.

Definition 2 A function is positive definite if, for all finite subsets and all complex coefficients ,

Clearly,

Usually, the cone of positive definite functions is much easier to describe.

3.2. A toy theorem

Here is a toy theorem that illustrates the power of the method, although it does not suffice to determine maximal densities.

Theorem 3 Let be a metric space. Let be a integer and . Suppose there exists a copositive function such that and

Then an -code in contains at most points.

Proof. Let be an -code. Then

hence . q.e.d.

So a hard task is to guess a clever function and show that it is copositive. The procedure usually works only in very symmetric situations, where turns out to be positive definite.

4. Success stories

Now I describe a few instances where the procedure works.

4.1. The Hamming code

We view the Hamming cube as a -dimensional vectorspace over the field with elements.

The Hamming code is a -dimensional subvectorspace of . It is an -code for . This is optimal, as can be proved using a positive definite function on (which turns out to be a polynomial).

4.2. The binary Golay code

is a -dimensional subvectorspace of . It is an -code for .

5. Back to Euclidean space

Here, the sought for function on will be a Schwartz function, i.e. it is infinitely differentiable, and it decays, as well as all its derivatives, faster that any power of . To such a function , we associate the radial function on .

Theorem 4 (Cohn-Elkies 2003) Suppose that there exists a Schwartz function such that

1. , .
2. the Fourier transform is nonnegative on .
3. .

Then each periodic configuration which is an -code must have density at most .

Note that the assumption is related to being positive definite.

Thus Theorem 4 allows to bound from above the densities of periodic packings. We have little hope to obtain a sharp bound. However, Theorem 4 is merely a generic statement, it can be refined, yielding sharp bounds.

In the same paper, Cohn-Elkies found numerically very good solutions in dimensions 8 and 24 :

where and Leech denote very regular periodic configurations which exist only in those dimensions.

6. Sharp results

My contribution is to replace the numerical constant by .

Theorem 5 (Viazovska 2016) There exists a Schwartz function that satisfies Cohn-Elkies’ assumptions in dimension for . This implies that the lattice has maximal density among all configurations (periodic or not) in .

Theorem 6 (Cohn-Kumar-Miller-Radchenko-Viazovska 2016) There exists a Schwartz function that satisfies Cohn-Elkies’ assumptions in dimension for . This implies that the Leech lattice has maximal density in .

6.1. Further remarks

The method provides upper bounds on in other dimensions, but it does not seem to provide sharp bounds in most dimensions (this statement can be made precise).

In 2 dimensions, numerical evidence indicates that there could exist a Schwartz function yielding the sharp bound, but one is unable to prove it yet.

Notes of Corinna Ulcigrai’s Hadamard Lectures, june 2022

Parabolic dynamics

1. Survey

1.1. What does hyperbolic, elliptic mean in dynamics?

Let be a dynamical system. Even if deterministic, it can exhibit a chaotic behaviour. This has several characteristics. One of them is the Butterfly effect: sensitive dependence on initial conditions (SDIC).

Definition 1 A flow has SDIC if there exists such that , , such that and such that .

A quantitative measurement of SDIC is provided by the dependence of on .

Definition 2 Let be a nondecreasing function. A flow has SDIC of order if in the above definition, one can take such that .

This leads us to a rough division of dynamical systems:

1. Elliptic: no SDIC, or if any, subpolynomial.
2. Hyperbolic: SDIC is fast, exponential.
3. Parabolic: SDIC is slow, subexponential.

This trichotomy is advertised in Katok-Hasselblatt’s book.

For instance, entropy is a measure of chaos. Elliptic or parabolic dynamical systems have zero entropy. Hyperbolic dynamical systems have positive entropy.

1.2. Examples of elliptic dynamical systems

1. Circle diffeomorphisms.
2. Linear flows on the torus (these two examples are related, one is the suspension of the other).
3. Billiards in convex domains. Usually, there are many periodic orbits, trapping regions, caustics.

This the realm of Hamiltonian dynamics and KAM theory.

1.3. Examples of hyperbolic dynamical systems

1. Automorphisms of a torus which are Anosov, i.e. all eigenvalues have absolute values . Then orbits diverge exponentially: if , , set . Then mod and reaches in time such that for , an exponential.
2. Geodesic flows on constant curvature surfaces.
3. Sinai’s billiard: a rectangle with a circular obstacle. Somewhat equivalent to the motion of two hard spheres on a torus. Scattering occurs after hitting the obstacle, due to its strict convexity.

This is the realm of Anosov-Sinai and others’ dynamics. Structural stability occurs: hyperbolic systems form an open set.

1.4. Examples of parabolic dynamical systems

1. Horocycle flows on constant curvature surfaces. They were introduced by Hedlund, followed by Dani, Furstenberg, Marcus, Ratner.
2. Nilflows on nilmanifolds. If is a cocompact lattive in a nilpotent Lie group, let be a -parameter subgroup of acting by right translations on . It descends to . Both examples are algebraic, this provides us with tools to study them. They are a bit too special to illustrate parabolic dynamics.
3. Smooth area-preserving flows on higher genus surfaces.
4. Ehrenfest’s billiard: rectangular, with a rectangular obstacle. Here, SDIC is only caused by discontinuities due to corners. More generally, billiards in rational polygons (angles belong to ), or equivalently linear flows on translation surfaces. This field is known as Teichmüller dynamics. Forni considers them as elliptic systems with singularities, I prefer to stress their parabolic character.

1.5. Uniformity

Within hyperbolic dynamics, there is a subdivision in uniformly hyperbolic, nonuniformly hyperbolic and partially hyperbolic.

In the same manner, we see horocycle flows as uniformly parabolic, and nilflows as partially parabolic, with both elliptic and parabolic directions. Area-preserving flows on surfaces have fixed points which introduce partially parabolic behaviour: shearing is uniform or not. However, there is no formal definition.

1.6. More examples of parabolic behaviours

Parabolicity is not stable. However, Ravotti has discovered a -parameter perturbation of unipotent flows in .

A flow is a time-change of a given flow if there exists a function such that

For to be a flow, it is necessary that be a cocycle.

Both flows have the same trajectories. A feature of parabolic dynamics is that a typical time-change is not isomorphic to and has new chaotic features. Indeed, an isomorphism would solve the cohomology equation, and there are obstructions.

1.7. Program

Study smooth time-changes of algebraic flows.

Goes back to Marcus in the 1970’s. Algebraic tools break down, softer methods are required: geometric mechanisms. Also, we expect the features exhibited by time changes to be more typical.

2. Chaotic properties

2.1. Definitions

Definition 3 Let be a measure space with finite measure. Let be a measure preserving flow. The trajectory of a point is equidistributed with respect to if for every smooth observable ,

tends to as tends to .

is ergodic if almost every has equidistributed orbit with respect to .

This is Boltzmann hypothesis.

Definition 4 Say is mixing if for all , the correlation

tends to as tends to .

This means decorrelation of functions. This implies ergodicity.

The speed at which decorrelation occurs is a significative feature too.

Definition 5 The speed of mixing is a function such that for all smooth observables , the correlation decays at speed , i.e.

as tends to .

2.2. Relation to the trichotomy

Elliptic systems often are not ergodic, but even when they are, they are not mixing.

Hyperbolic and parabolic systems can be mixing, but at different speeds:

• hyperbolic exponential decay of correlations.
• in parabolic systems, we expect that, if mixing occurs, the decay of correlations is slower: polynomial or subpolynomial.

A related concept is that of polynomial deviations of ergodic averages: if is smooth and has vanishing integral, s ergodic and is an equidistribution point,

and no faster. This phenomenon was first discovered on horocycle flows, then Teichmüller flows (Zorich, experimentally, Kontsevitch-Zorich for a proof).

2.3. Other features

Spectral properties. Let be the operator on . Then is unitary. What is its spectrum?

Disjointness of rescalings. Rescaling means linear time change .

3. Results

Horocycle flow is mixing (Ratner). The spectrum is Lebesgue absolutely continuous.

Time changes of horocycle flows are mixing (Marcus, by shearing). This can be made quantitative (Forni-Ulcigrai).

Disjointness of rescalings fails for horocycle flows, but hold for nontrivial time changes (Kanigowski-Ulcigrai and Flaminio-Forni). The spectrum is Lebesgue absolutely continuous as well.

Nilflows themselves are not mixing, but typical time changes of nilflows are mixing (Avila-Forni-Ravotti-Ulcigrai).

We shall see geometric mechanisms at work:

• Mixing via shearing.
• Ratner property of shearing.
• Renormizable parabolic flows.
• Deviations of ergodic integrals.

4. Horocycle flows

Today’s goal is to explain the technique of mixing by shearing.

4.1. Algebraic viewpoint

Consider and its subgroups

Every matrix can be uniquely written , hence . The factors do not commute, because of the key relation

The key relation can be interpreted as a selfsimilarity property: is a fixed point of renormalization by .

Take a discrete and cocompact subgroup . Then and act on by left multiplication. The key relation implies that the rescaled flow $latex {h_{\mathbb R}^k=(h_{ks})_{s\in{\mathbb R}}}&fg=000000$ is conjugated to $latex {h_{\mathbb R}}&fg=000000$. This fails for other (nonrescaling) time changes, as I proved recently with Fraczek and Kanigowski.

4.2. Geometric viewpoint

Let denote the upper half plane, with metric .

Fact. acts isometrically and transitively on , and this yields a diffeomorphism of with the unit tangent bundle .

Indeed, the action is by Möbius transformations

on , and by their derivatives on .

With this identification, orbits of are curves which project to geodesics of , and coincide with their lifts by their unit speed vector. On the other hand, orbits of are curves which projects to horocycles of , and coincide with their lifts by their unit normal outward pointing vectors. Lifts by inward pointing normal vectors are orbits of .

is a hyperbolic flow: it contracts in the direction of -orbits, it dilates in the direction of -orbits,

4.3. Classical results

Let denote Haar measure on . It maps via to hyperbolic volume. Consider the induced measure on (still denoted by ). Then and acts on by measure preserving transformations.

Then

1. is ergodic (Hopf). It is far from being unique ergodic (plenty of periodic orbits).
2. is mixing.
3. is uniquely ergodic (Furstenberg). This means that every orbit is equidistributed with respect to .

5. Shearing

This is an alternate way to prove mixing. The idea goes back to Marcus (Annals of Math. 1977). Marcus covered a more general situation, and proved mixing of all orders (i.e. for multicorrelations, integrals involving an arbitrarily large number of functions).

Let us shift viewpoint on the key relation. Let be a piece of -orbit of length . Let

Then is sheared or tilted in the direction of the geodesic flow.

Key idea in parabolic dynamics: In several parabolic systems, the Butterfly effect happens in a special way, e.g. shearing. Points nearby move parallel, but with different speeds. This implies that transverse arcs shear.

5.1. Recipe for mixing in parabolic dynamics

Here are the ingredients:

• A uniquely ergodic flow .
• A transverse direction which is sheared in the direction of the flow.

By assumption, for every , the trajectory equidistributes with respect to . We want to upgrade it to mixing, which is a property of sets: indeed

is equivalent to

i.e equidistributes as .

The idea is to cover by short arcs in the transverse direction. We prove that each such arc equidistributes, and apply Fubini. I.e. if , apply . Then .

Each becomes close to a long piece of orbit of . By unique ergodicity, that piece equidistributes, and this implies equidistribution for .

5.2. Time change

Let be a smooth time change, which is a cocycle with respect to a given smooth flow , i.e.

We are interested in the flow defined by

The generator of is

It is a smooth nonnegative function on . We assume that . We denote by .

Remark. If is generated by a smooth vectorfield , then is generated by the vectorfield .

Theorem 6 (Forni-Ulcigrai) Let be the horocyclic flow of a compact constant curvature surface. For any smooth function , the flow is mixing, with quantitative estimates which imply that the spectrum is absolutely continuous with respect to Lebesgue measure.

Lemma 7 Let denote the generator of and the generator of . Take a segment of -orbit

Let Then where

Remark. , hence . Thus, as tends to , tends to a finite limit, the shear rate.

Proof of Lemma. It relies on , which implies that .

We see that we need compute integrals over sheared arcs . Let be a smooth function. Then

The second term is an ergodic integral which is easy to handle. The main term is the first term, which can be rewritten

where , .

To deduce mixing, one must estimate inner products . We integrate by parts

The second term is again an ergodic integral that tends to .

5.3. Quantitative equidistribution estimates

We are interested in ergodic integrals of the form

Flaminio-Forni treat the un-time-changed case and show that

for some . One can adapt their arguments, using estimates by Bufetov-Forni, to the time-changed case, and get similar estimates.

5.4. Additional references

Marcus original technique already proved mixing. His setting was Anosov flows, with their stable and unstable foliations. From these, a flow can be defined, which satisfies

where $latex {s^*}&fg=000000$ has a continuous mixed partial second derivative $latex {\frac{\partial s^*}{\partial t \partial s}}&fg=000000$. So we see that time-changes were already in the picture.

Kushnirenko was able to prove mixing for smooth time-changes, assuming

Thus small time-changes are mixing. What about larger ones? This is still open.

Tiedra de Aldecoa uses a different method to prove absolute continuity of the spectrum for time changes satisfying (KC).

Generalizations. The setting is algebraic dynamics: a unipotent -parameter subgroup acting on , semisimple Lie group.

Lucia Simonelli (Forni’s student) could prove absolute continuity of the spectrum for time changes satisfying (KC).

Davide Ravotti (my student) could prove quantitative mixing.

Kanigowski and Ravotti could prove quantitative -mixing.

6. Heisenberg nilfows

Let denote the Lie group of unipotent matrices, with as standard generators of its Lie algebra, . Let be a discrete cocompact lattice (for instance, unipotent matrices with integer entries). We call the Heisenberg nilmanifold. acts on by right multiplication. The action of a -parameter subgroup is called a nilflow.

6.1. Classical results

Auslander-Green-Hahn (1963) studied unique ergodicity of nilflows. They showed that is can be written , for the nilflow defined by ,

unique ergodicity ergodicity minimality and are rationally independent.

Rational independence means that no linear relation with nonzero integral coefficients can hold.

In other words, if we project the situation to the -torus where , , then the flow projects to a flow on , and

unique ergodicity for unique ergodicity for .

Here, we have used a theorem of Furstenberg on skew-products of rotations of the circle.

Definition 8 For real numbers , let be the diffeomorphism of the -torus defined by

In general, a skew-product over a map is a map which is fiber-preserving (with respect to the projection ) and the permutation of fibers is given by . In the example at hand, is isometric on fibers.

6.2. First return map

Lemma 9 Assume that the Heisenberg nilflow is uniquely ergodic. There is a transverse submanifold , diffeomorphic to a torus, such that the Poincaré return map , given by , the first return time to , is one of the Furstenberg skew-products .

Proof of the Lemma. lifts to a vertical plane in . Since and commute, is diffeomorphic to a torus. If is uniquely ergodic, , so is transverse to the flow.

We show that is a return time. We use the fact that . So using the Campbell-Hausdorff-Dynkin formula, we compute

This point belongs to , so .

6.3. Lack of mixing

The above Lemma shows th at we can now focus on Furstenberg skew-products. We shall see that the parameter plays no role, so we focus on .

Definition 10 Given a map and a function (called the roof function), we define the special flow over under as follows: it is a flow on an -bundle over the circle, the quotient of the vertical unit speed flow on under the identification

Fact. If a flow admits a global Poincaré section with first return time , then is isomorphic to the special flow of the first return map with roof function .

In the case at hand, the roof function is constant. Therefore, the special flow is not mixing: if , a short interval, so do its images by the vertical unit speed flow, and so do their projections to , which are the images of a set and its images by the special flow.

This shows that Heisenberg nilflows are never mixing.

7. Mixing time-changes

The following contents can be found in Avila-Forni-Ulcigrai. A recent generalization to all step 2 nilflows can be found in Avila-Forni-Ravotti-Ulcigrai. Ravotti has treated filiform nilflows.

Let be a Heisenberg nilflow and a smooth positive function on . Let

Then the time-change is given by

7.1. Time-changes versus special flows

We have seen that Heisenberg nilflows are special flows with constant roof function. The time-change is again a special flow, over the same skew-product, but with roof function

7.2. Trivial time-changes

Beware that there exist smooth time-changes which are trivial, i.e. smoothly conjugate to the original nilflow.

In general, adding a coboundary to the roof function of a special flow produces an isomorphic flow. This leads us to the following problem: understand cohomology of nilflows. Here are our ultimate results.

Theorem 11 Let be a There exists a dense set in of roof functions, and a vectorspace of countable dimension and codimension, such that if a roof function is chosen in , the corresponding special flow is mixing.

Moreover, for ,

In fact, Katok has found a nice characterization of which are smoothly trivial. I will come back to this next week. Today, I merely give one example.

Example. is a smoothly trivial roof function.

Under the assumption that has bounded type, Kanigowski and Forni have proved quantitative mixing.

7.3. Idea of proof

We start from a Furstenberg skew-product and play with roof functions. We use again mixing by shearing. We consider intervals in the fiber (i.e. in the direction). We shall see that many of them shear in the flow () direction. But there are intervals which do not shear or shear in the other direction.

7.4. Special flow dynamics

Given a point in the torus and , we compute . When is large, we join bottom to roof several times. Let

and

Then

In order to exhibit shearing, we want to see how this changes in . Since is an isometry in the -direction, the -derivative of the sum is the sum of -derivatives, i.e.

Take trigonometric polynomials on the torus, which are positive.

Given , let minus its average on the -fiber. Define as the set of such that is not a measurable coboundary.

Step 1. Since is not a measurable coboundary, the sums must grow,

This relies on a result by Gottschalk-Hedlund, plus decoupling.

Step 2. The sums are trigonometric polynomials of bounded degree. It follows that

There can be flat intervals where no stretch occur, so one must throw them away. But the larger , the shorter these intervals are.

Step 3. We use a polynomial bound on level sets of trigonometric polynomials.

The next class will deal with renormalization, deviations of ergodic averages. Only later shall we get back to Katok’s characterization of smoothly trivial special flows and to nilflows.

8. Short recap

Parabolicity is (a bit heuristically) defined by slow butterfly effect. This can be formalized for smooth flows, in terms of growth of derivatives under iteration. It is not that easy to build examples.

Presently, parabolic flows is the following list of examples,

• Horocycle flows of compact constant curvature surfaces.
• Unipotent flows (a generalization of the above).
• Nilflows and their time-changes.
• Smooth area-preserving flows.
• Linear flows on flat surfaces with conical singularites.

The two first are uniformly parabolic. The next is partially parabolic. The fourth is nonuniformly parabolic, the last is elliptic with singularities.

We have proven mixing via the technique of shearing.

Here is a further example where this technique works, due to B. Fayad, of an elliptic flavour. Start with a linear flow on the -torus. If , Fayad has been able to construct anaytic time-changes which are mixing.

Such examples are very rare (they rely on parameters being very Liouville numbers).

9. Renormalization

The word here is taken in a meaning which differs from its use in holomorphic dynamics (not to speak of quantum mechanics).

The idea is to analyze systems which are approximately self-similar, and exhibit several time scales.

We introduce the renormalization flow which rescales: long trajectories become short. Given a map , one way to zoom in is to restrict to a subspace and replace with the first return map to . Eventually rescale space afterwards. But there are other means.

9.1. A series of examples

Example. Start with the horocycle flow . The key relation is

Applying the geodesic flow to a length trajectory , we get a trajectory of length . So the geodesic flow achieves renormalization, on the same space, with no effort.

Example. The cat map associated with the matrix on the -torus. Let , denote the eigenvalues, the eigenvectors. Let be the linear flow in direction , with unit speed. This is an elliptic flow.

Put . Let

Put . Then maps to a trajectory of of length .

Express as the composition of two Dehn twists,

Rotate coordinates so that eigenvector becomes vertical. Then acts by a diagonal matrix in , which was denoted by , earlier.

Example. Higher genus toy model.

View a genus surface as a regular octagon in the Euclidean plane with edge identifications. Fix a direction , consider the linear flow in direction . It descends to a flow on the surface which is not well-defined at the vertex

This generalizes to translation surfaces, made of a disjoint union of polygons, with identifications given by translations. The notion of a direction is well-defined on the surface (except at finally many singularities), whence a flow with singularities.

Consider matrix

It applies a shear on the original regular octagon. Since , This linear map induces a homeomorphism of the surface . Let us repeat in different direction (rotation by 45 degrees), get matrix . Let , this is a hyperbolic matrix with eigenvalues . We get again an affine automorphism of the surface. Let denote the linear flow in the direction of the eigenvector . Then

B. Veech has shown that the affine group of the surface is generated by and the order rotation. This is as large as the affine group of a translation surface can be. Therefore, is clled the Veech surface.

For almost every direction (the condition is that ), there exists a sequence of hyperbolic automorphisms with unstable direction that converge to . can be used to renormalize .

Example.

For a slightly deformed octagon , the affine group of the corresponding surface is trivial. Nevertheless, almost every linear flow is still renormalizable: there is a sequence of surfaces and affine hyperbolic morphisms with expanding direction converging to .

Indeed, consider the space of linear flows on translation surfaces of genus . The flow of diagonal matrices acts on this space, generating a flow on , known as the Teichmüller flow.

Theorem 12 (Masur-Veech) is recurrent.

Therefore for almost every linear flow, there exists a sequance such that tends to .

9.2. What is renormalization good for?

It is used to put diophantine conditions on linear flows in higher genus. See my ICM 2022 talk (watch it on line on july 11th). I do not pursue this topic further.

It is used to study deviations of ergodic averages. Assume a flow is uniquely ergodic. Ergodic integrals take the form

By the ergodic theorem, for every . Let us focus on our favourite example, the Veech surface. The invariant measure is area in the plane. Unique ergodicity is a theorem of Masur, Kerckhoff-Masur-Smillie

We show that for functions with vanishing average,

for some (discovered by Zorich, conjectured by Kontsevitch-Zorich, proven by Forni).

Fix a basis of . Fix a section of the linear flow. Take trajectories from and to their first return to . Closing them by segments of sigma gives representatives of the basis of homology. Let

Let be the matrix expressing the homology basis is the initial basis. Let denote the eigenvalues of . By continuity of , up to a small error,

Let us start from a large instead of ,

For simplicity, let us assume that is the eigenvector . Then

a contradiction. Therefore must be a combination of . Thus

On the other hand, grows like . Let satisfy

Then , as announced.

Of course, I cheated a bit, some more regularity of is needed.

In that example, the fact that can be explicitly. Tomorrow, I will mention results on this for general translation surfaces.

And then, back to nilflows.

9.3. Area preserving flows on surfaces

compact connected oriented surface of genus . Let be a flow which preserves a smooth measure . Note that has fixed points, which are all of saddle type. In the next theorem, the number and types of fixed points are fixed.

Theorem 13 (Zorich, Forni, Avila-Viana) There exists distinct positive exponents such that almost every choice of (Katok’s fundamental class, described in terms of periods) is uniquely ergodic, and for all smooth functions ,

for all .

In this statement, means a function such that

Furthermore, is a distribution on , and .

This type of behavior is called a power deviation spectrum. It was conjectured by Kontsevitch and Zorich, then proved by Zorich in 1997 for special functions , with only one term . Then Forni obtained a proof in 2002 for functions with support away from fixed points, up to distinctness of exponents which was proven by Avila-Viana. Bufetov gave a more precise version, transforming the result into an asymptotic expansion. With Fraczek, we gave a different proof based on Marmi-Moussa-Yoccoz). Finally, Fraczek-Kim could handle generic saddles. The expansion then involves extra terms depending on saddles (and not on ).

9.4. Idea of proof

Choose coordinates such that appears as a time-change of a linear flow on a translation surface. The time change is smooth only away from fixed points. If function has support away from fixed points, one is reduced to study deviation for linear flows. For a general linear flow, a renormalization is given by a sequence of matrices , the Kontsevitch-Zorich cocycle. Eigenvalues are replaced with ratios of Lyapunov exponents.

10. More on renormalization

10.1. Back to nilflows

Let denote the stabilizer of vector is . Defined by

Then is recurrent, its has been used as a renormalization by Flaminio-Forni, in order to prove polynomial deviations of ergodic averages.

10.2. Renormalizable parabolic flows

Horocycle flows: yes.

Unipotent flows: unknown.

Heisenberg nilflows: yes.

Higher step nilflows: unknown in general. Special case (special flows over skew products) studied by Flaminio-Forni. In that case, the maps diverge.

Linear flows over higher genus surfaces (they are smooth and area-preserving).

More general smooth flows on surfaces (not necessarily area preserving). Then typically diverges. This is related to generalized interval exchange transformations.

11. Isomorphisms between time-changes

Recall that a time-change of a flow is

When do such a change lead to a genuinely different, nonisomorphic flow?

11.1. Setting of special flows

Let be a map. Given a roof function $latex {\Phi:Y\rightarrow{\mathbb R}_{>0}}&fg=000000$, the flow of translations on descends to a flow $latex {\psi^{f,\Phi}_{\mathbb R}}&fg=000000$ on the quotient space

Lemma 14 Let be special flows over the same map , under roofs and . If there exists a function such that

then the two special flows are isomorphic.

Indeed, look for a conjugating homeomorphism of the form

It commutes with translations and maps one equivalence relation to the other.

11.2. Cohomological equations

The operator is called the coboundary operator. Hence the equation with unknown is called a cohomological equation. It sometimes appears with a twist: , for some .

There are obvious obstructions.

If is a periodic point, i.e. , then

So every periodic orbit gives an obstruction. If is hyperbolic, it is essentially the only one.

More generally, if is an invariant measure, then

So every invariant measure gives an obstruction.

11.3. An elliptic example

If is elliptic, this is sometimes sufficient, e.g. for circle rotations : if is a trigonometric polynomial and , then there exists a solution . If is smooth, a Diophantine condition on is required in addition for the solution to be smooth. Indeed, in Fourier, if

then

For to decay superpolynomially, one needs that

for some , which amounts to being badly approximable by rationals.

11.4. Parabolic case

In the parabolic world, in addition to invariant measures, invariant distributions provide further obstructions.

Remember that Heisenberg nilflows are special flows with constant roof over Furstenberg skew-products of the form

We need study the corresponding cohomological equation.

Proposition 15 In Fourier series, if

then the cohomological equation has a formal solution if and only if all where is the distribution such that

Indeed, let , and

Then

The matrix acts on . It has a family of orbits and orbits

, for each . splits accordingly, so the cohomological equation can be solved independently in each summand

and on

For , acts like the circle rotation , so we already understand the necessary condition, given by invariant measures.

For fixed and , let us denote

The cohomological equation reads

where

Recursively, one gets

Then

Similarly, the equation

yields

Combining both leads to

i.e. .

Conversely, one can see that this conditions are sufficient for existence of a formal solution, and for existence of smooth solutions under Diophantine conditions.

11.5. More general results on cohomological equations in parabolic dynamics

• The case of Heisenberg nilflows, which we just treated, is due to Katok.
• More general nilflows have been studied by Flaminio-Forni, as well as horocycle flows.
• Linear flows on higher genus surfaces are due to Forni. In this case, one gets distributions . If a smooth function is killed by all of them, ergodic integrals stay bounded. This is equivalent to being a coboundary, according to Gottschalk-Hedlund.

11.6. Cocycle effectiveness

Sometimes, the cohomological equation with measurable data is needed. It is much more difficult, but some miracle occurs in the parabolic setting.

Definition 16 Given a map , say a function on is a measurable coboundary of the exists a measurable such that .

The theorem on mixing smooth time-changes of Heisenberg nilflows (Avila-Forni-Ulcigrai) required the roof not to be a measurable coboundary. It turns out that here, this is equivalent to not being a smooth coboundary.

Proposition 17 Let us study the Furstenberg skew-product on the -torus. Let be a smooth function on . Then
is not a smooth coboundary is not a measurable coboundary.

Indeed, let

denote the Birkhoff sums. Then Flaminio-Forni establish quadratic upper bounds: there exists a sequence such that

Matching lower bounds exist: if is not a smooth coboundary, there exists a nonvanishing , and

If is a measurable coboundary, , then

stays bounded on a set of almost full measure, and grows at most quadratically on the complement. This contradicts the quadratic lower bound.

11.7. More general nilflows

Theorem 18 (Avila-Forni-Ravotti-Ulcigrai) For general nilflows of step , there exists a dense (in ) class of generators of time-changes, which are

• either measurably trivial (i.e. measurably conjugate to the nilflow);
• or mixing.

Unfortunately, the set is not explicitly describable like in the Heisenberg case.

The proof is an induction on central extensions. It uses mixing by shearing.

12. More examples of parabolic dynamics

12.1. Parabolic perturbations which are not time-changes

These were discovered by Ravotti during his PhD at Princeton.

Here, parabolic means that the derivative of the flow grows polynomially. It implies that smooth time-changes are still parabolic.

Start with , a cocompact lattice, . Let be the flow generated by a unipotent element . Let belong to the center of the minimal unipotent and such that .

Let where is a function on such that . Let denote the corresponding flow.

Theorem 19 If preserves a smooth measure with density, then is

1. parabolic: ;
2. ergodic;
3. mixing.

Remark. Existence of there exists a time-change of the -flow which commutes with .

Remark. There exists such which are not smoothly isomorphic to . This follows from the failure of cocycle rigidity for parabolic actions, due to Wang, following many people.

Remark. Ergodicity needs be proven, it does not follow from general principles.

The proof relies on mixing by shearing, although in a setting different from what we have already met. Consider arcs of orbits of and push them by . Since

where is an ergodic integral for .

12.2. What else can shearing be used for?

A strong, quantitative, shearing can be used to establish spectral results. Here, I mean the spectrum of the Koopman operator

To each , there corresponds a spectral measure on . Its Fourier coefficients are given by selfcorrelations of , i.e.

The spectrum of is absolutely continuous for all , is absolutely continuous for all , .

If is ergodic, it is enough to study functions which are smooth coboundaries.

Theorem 20 (Forni-Ulcigrai) Smooth time-changes of a horocycle flow has absolutely continuous spectrum.

The proof uses quantitative bounds

Since , this implies absolute continuity.

Fayad-Forni-Kanigowski consider smooth area-preserving flows on the -torus with a stopping point.

12.3. Ratner property

M. Ratner uses a quantitative form of shearing for unipotent flows.

Shearing takes some time. Ratner requires the following:

For all , for all large enough , there exists a set of measure , for all pairs not in the same orbit, but such that , there exists such that

and

for all .

For a long time, this was used only in algebraic dynamics, until a more flexible variant, called switchability, was introduced. It means that one can switch past and future.

Fayad-Kanigowski and Kanigowski-Kulaga-Ulcigrai established this variant for typical smooth area-preserving flows on surfaces. This implies mixing of all orders.

Here is another application if these ideas:

Theorem 21 (Kanigowski-Lemanczyk-Ulcigrai) For all smooth time-changes of the horocycle flow, the rescaled flow is not isomorphic to .

They are actually disjoint in Furstenberg’s sense. Recall that the horocycle flow itself is isomorphic to its rescalings.

We use a disjointness criterion based on this switchable variant of Ratner’s property.

12.4. Summary

• Parabolic means slow butterfly effect.
• Typically slow mixing.
• Slow equidistribution.
• Disjointness of rescalings.
• Obstructions to cohomological equation.

Tools:

• Shearing.
• Ratner property and switchability.
• Renormalization.

Notes of Uri Bader IHES lectures october 5th, 2021

Algebraic representations of ergodic actions

Based on joint work with Alex Furman, and earlier litterature.

Thanks to Sami Douba for his help with notetaking.

Today, we start with basics on measure spaces and algebraic varieties. Later on, we shall merge both subjects together.

1. Ergodic theory of algebraic varieties

1.1. Algebraic actions

is a local field. For simplicity, characteristic is but most of what I will say extends to positive characteristic. Also to nonlocal complete normed fields. is an algebraic group acting algebraically on an algebraic variety . Then acts on . We equip with the -topology, it is second countable and locally compact.

Questions. What can one say of the structure of orbits?

Examples. acting on has orbits, one open and one closed. Let , triangular matrices, diagonal matrices, unipotent matrices. Then -orbits are concentric circles, -orbits are the origin, two halflines and two halfplanes. -orbits are the origin, four halflines and branches of hyperbolas. -orbits are points of the -axis and two halplanes.

We observe that orbits are open or closed, except for halflines which are nearly closed: intersections of an open and a closed set. We call such sets locally closed

Fact: Orbits of algebraic actions are locally closed.

Consequences.

-invariant open sets separate points in distinct orbit in . It follows that the quotient topology on is second countable and (topology separates points). This is known as Chevalley Theorem (combined with a result of Borel-Serre).

The Borel structure on is countably separated. It follows that there exists a Borel embedding of to .

A fundamental theorem of Descriptive Set Theory states that all uncountable Polish topological spaces (completely metrizable, admitting a countable dense subset) are isomorphic as Borel spaces. See Kechris’ book. Such Borel spaces are called standard Borel spaces.

Examples. Finite sets, countable sets, a separable Hilbert space.

A Borel space is said to be countably separated if there exists a countable collection of Borel sets that separates points (equivalently, space has a Borel embedding into ). Standard Borel sets have this property.

We take this encouraging fact as an invitation to do ergodic theory.

1.2. measures and measure classes

Say two measures on a Borel space are equivalent if they have the same sigma-ideal of null sets. A measure class could be understood as the choice of a sigma-ideal.

Warning. Not every sigma-ideal comes from a measure. For instance, the sigma-ideal of meager sets does not arise from a measure.

Examples.

has a natural volume measure class.

has the Haar measure class.

For every closed subgroup , has a unique -invariant measure class, called the Haar class. Warning: very rarely does admit a -invariant measure. For instance, acting on projective space has no invariant measure in the Haar class. More generally, when is simple and is parabolic, has no invariant measure in the Haar class.

If is a locally compact group, Haar measure is finite iff is compact. If is a closed normal subgroup, then has a finite invariant measure iff is cocompact in .

Definition 1 Say that a -invariant measure class on is ergodic if every -invariant Borel set is either null or full (complement is null).

Equivalently, every a.e. defined Borel -invariant map is a.e. constant. Here, can be replaced with any countably separated space.

Corollary 2 Every -ergodic measure class on is supported on a single orbit. Moreover, it coincides with the Haar class on this orbit.

Indeed, think of the action as a -map where is a -ergodic space. Since is countably separated, this map must be a.e. constant. This leads to a map to an orbit. This must be a Borel and measure class isomorphism, thanks to the uniqueness of the Haar class.

In the nonergodic case, one can use ergodic decompositions.

Definition 3 Given a -measure class on , since is countably separated, for every measure in the class, there exists a family of -invariant measures on , , such that is a Haar measure on the orbit denoted by , and

This is the ergodic decomposition of .

In particular, the measure class is fully determined by the class of the pushed-forward measure .

1.3. Classification of -invariant probability measures

If is a normal -algebraic subgroup, which is cocompact, then has a finite Haar measure. By Noetherianity, there exists a minimal element among such cocompact normal -algebraic subgroups. In fact, is a least element. Indeed, given and , maps to a closed subset of hence is compact.

Theorem 4 If has no compact factors (i.e. ), then every -invariant probability measure on is supported on fixed points.
In general, every -invariant probability measure on is supported on the -fixed points.

Example. For acting on , the only -invariant probability measure is the Dirac mass at the origin. Same picture for , and (with as many invariant measures as there are measure on the -axis). However, has a lot of invariant measures.

This implies Borel’s density theorem.

Theorem 5 (Borel) If has no compact factor, and is a lattice, then is Zariski-dense in .

Indeed, let be the Zariski closure of , then push the -invariant probability measure from to . It must be supported on a fixed point, i.e. is a point.

1.4. Generalization

Fix a locally compact sigma-compact group and a representation . Then acts on . The orbit space is complicated, but still is a -map. Same reasoning yields: every -invariant ergodic measure is supported on a unique -orbit.

Theorem 6 There exists a minimal normal -subgroup such that

has precompact image.
Every -invariant measure on a -algebraic variety is supported on the -fixed points.

Consider , for .

Corollary 7 The stabilizer of a measure is compact modulo the fixator of the Zariski-support of .

Fact (Zimmer). The action of on has locally closed leaves.

1.5. Some more ergodic theory

A Lebesgue space is a standard Borel space equipped with a measure class.

By a map from a Lebesgue space to a Borel space , we mean an equivalence class of Borel maps : defined almost everywhere, up to almost everywhere equality. The space of such maps is denoted by .

A morphism of Lebesgue spaces is a map which sends null sets to null sets.

Let be a locally compact second countable group. Then has a standard Borel space structure and a Lebesgue space structure. An -Lebesgue space is a Lebesgue space with a homomorphism such that is a morphism.

An action of on is ergodic if every -invariant map is essentially constant, for every standard Borel space .

Definition 8 Fix an action of on . Say it is

1. Doubly ergodic if the diagonal action on is ergodic.
2. metrically ergodic if for every isometric action of on a separable metric space , every -equivariant map is (essentially) constant.
3. Weakly mixing if for every ergodic probability measure preserving action of on , the diagonal action on is ergodic.
4. Has no compact factors if for every continuous homomorphism to a compact group and any compact subgroup , for every map (equipped with Haar measure), .

Easy fact. .

Indeed, if is doubly ergodic and acts isometrically on , the distance defines an invariant function on , hence constant. If the constant is not zero, the ilage of is discrete in , hence countable (since is separable), contradiction.

If is metrically ergodic and has a pmp action on , an invariant function on gives rise to an equivariant map . Since probability measure is invariant, is equivariant and the action on is isometric. Now is separable, so is constant.

If is weakly mixing and is a compact factor, one can assume . Take . The map is -equivariant , hence constant, so is a single point.

Easy fact. If the action of on is probability measure preserving, then .

Indeed, it suffices to prove that . One can assume that metric space has an invariant probability measure , fully supported, and that is complete. One easily shows that is compact. Then is compact. It must act transitively on , . Under assumption 4, is a point, this is .

1.6. metric ergodicity

Nonexample. Let be a compact subgroup. Let be a homomorphism. Then the action of on is not metrically ergodic.

Example. Let be a countable group. Let be a probability space. Then the shift action of on is metrically ergodic.

Indeed, , and the shift action on is ergodic.

Claim. Let be a noncompact -simple algebraic group. Let be a lattice, a noncompact closed subgroup. Then the action of on and the action of on is metrically ergodic.

Indeed, metric ergodicity passes to lattices (pass from to ). For the action, this follows from Howe-Moore. Indeed, the action of on the pmp space is mixing. This implies decay of coefficients. Their restrictions to any closed noncompact subgroup decay, hence the mixing action of .

1.7. Amenability

Definition 9 (Zimmer) The action of on is amenable if there exists an -equivariant conditional expectation

Note that amenability implies the following weaker “baby amenability”, which is often used: for every compact convex -space , there exists an -map .

Example. If is an amenable subgroup, the action of on is amenable.

Fact. For every locally compact second countable group , there exists an action of on some Lebesgue space which is both amenable and metrically ergodic (the Furstenberg boundary).

Example. Let be a noncompact -simple algebraic group. Let be a lattice, let be a noncompact amenable subgroup. Then the action of on is both amenable and metrically ergodic.

2. Algebraic representations of ergodic actions

Now we merge algebraic groups and ergodic actions. Fix a locally compact second countable group , an action of on a Lebesgue space , a local field and an algebraic -group , . Fix a continuous homomorphism .

Definition 10 An algebraic representation of the action of on with respect to is a –-variety and an -equivariant map .
A morphism between to such AREAs and is a –-morphism such that .

Example. Let be a closed subgroup. Consider the action of on . Every algebraic representation of this action is given by a pair and a point in which is fixed by the Zariski closure of in .

If is Zariski dense, we get a map from to , where

We get an AREA , and for every AREA , we have a unique morphism such that .
In other words, is an initial object in the category of AREAs of . This holds in general.

Theorem 11 Let act ergodically on . Then there exists an initial object in the category of AREAs associated with , of the form

We think of an ergodic action of as a generalization of a closed subgroup, up to conjugacy. The theorem states that the initial object is indeed a Zariski-closed subgroup.

Proof of Theorem 11. Consider the set of -algebraic subgroups of such that there exists a, AREA . This is nonempty. By Noetherianity, one can pick a minimal element (it will turn out to be a minimum, up to conjugacy, but it is harder). We show that the map is an initial object.

Consider an other AREA . Consider the diagonal representation

By ergodicity, the image of lies in one single -orbit . Composing with projection, we get a -map , hence an embedding up to conjugacy. By minimality, . The other projection provides us with a -map , which is unique.

Theorem 12 Assume the action of on is pmp, is -simple and is unbounded (i.e. not contained in a compact subgroup). Then the initial object is trivial: any representation of is constant. Indeed, an -invariant probability measure on exists only if is precompact.

2.1. Consequences

Theorem 13 (Bader-Furman-Gorodnik-Weiss) Let be a noncompact -simple algebraic group. Let be a lattice, let be a noncompact -algebraic subgroup. Consider the -action on . For the inclusion of into , the initial object is the identity .

Application. Every Borel map which commutes with is a homothety.

Indeed, let be a -map. Let be the composition of with . Set

Then is right--invariant and left--invariant. Since the action of on is pmp and weakly mixing, is constant. Thus there exists such that , , i.e. is a -map.

Theorem 14 Let be an amenable and metrically ergodic -space. Let be a -simple algebraic group. Let be an unbounded homomorphism. Then there exists an initial object where is a proper subgroup.

Indeed, let be a parabolic subgroup of . Since acts on the convex space , by amenability, there exists an -map . Since -orbits in are locally closed, the image of the -map is contained in a single orbit , where is the stabilizer of a measure . By the structure theorem on measure stabilizers, the fixator of the Zariski hull of the support of is cocompact in . Up to conjugacy, it is contained in .

Assume that is trivial. Then is compact, acts by isometries on the separable space . By metric ergodicity, the -map is constant, which contradicts the assumption that is unbounded in .

Therefore, is not normal. is contained in the normalizer of in , which is a proper -algebraic subgroup of . The composition is a nontrivial AREA for with respect to .

3. Lattices in products

Today, we are aiming at rigidity results for lattices. Before entering the subject, let me sum up where we had reached last time.

3.1. AREAs continued

The tension between ergodicity and the very simple structure of algebraic actions creates an initial object in the category of AREAs. We call gate the initial object, because it is our entrance gate into the algebraic world.

Two theorems:

1. Theorem 12. For a pmp and metrically ergodic action, the gate is trivial.
2. Theorem 14. For amenable and metrically ergodic actions, the gate is nontrivial.

Remark: unbounded amenable subgroups of are not Zariski-dense.

3.2. Leftover from last time : functoriality

Proposition 15 Fix . The gate defines a functor from the category of -ergodic actions and the category of -algebraic -(coset)-varieties.

3.3. Introduction to lattices in products

Examples.

is a lattice in .

is a lattice in .

is a lattice in .

Definition 16 A lattice in a product is irreducible if its projections to both factors are dense subgroups.
Equivalently, the action of on each factor is ergodic.

Equivalently, the actions of and of on are ergodic.

Equivalently, the action of each on is ergodic.

Indeed, the action of on equipped with the weak topology is continuous.

3.4. Commensurability

Assume that is totally disconnected locally compact. Then there exists a compact open subgroup . Any two are commensurable. Say a subgroup is commensurated if for all , and are commen surable.

Let be an irreducible lattice. Let be a compact open subgroup. Then

is commensurated in , it is a lattice in . Hence is a lattice which is commensurated by the dense subgroup .
Conversely, assume that where is a lattice, is commensurated and is dense. Then one can reconstruct from these data. There exists a totally disconnected group , a dense embedding , and a precompact embedding such that is an irreducible lattice. It is called the Schlichting completion of .

This indicates that lattices in products Lietdlc are simpler that lattices in Lie groups, in the sens that we have a dual way of looking at them.

3.5. Superrigidity

Theorem 17 Let be an irreducible lattice. Let be a -simple group, let is Zariski dense and unbounded. Then superrigidity holds: extends uniquely to a continuous homomorphism and factors through one of the factors.

Corollary 18 If are as above (i.e. is commensurated and is dense), is -simple and is Zariski dense and unbounded on , then extends uniquely to a continuous homomorphism .

Indeed, the case where extends to is excluded.

3.6. Application to arithmeticity

Apply previous theorem to a lattice in and conclude that a lattice is arithmetic iff it has a dense commensurator.

3.7. Preparation for the proof

Fix an action of on a Lebesgue space which is amenable and metrically ergodic.

Claim. The diagonal action of on is amenable and metrically ergodic.

Indeed, assume is a nonempty -compact convex space. Then is nonempty, it is an -compact convex space (viewed as a subset of ). Therefore, there exists an -map . I.e., there exists an -map . This proves amenability.

Let be an -isometric metric space. Any -map is a.e. independant on the variable, and on the variable, therefore a.e. constant.

Corollary 19 For every irreducible lattice in , the action of on is amenable and metrically ergodic.

Of course, the action of on is not ergodic, it is proper, but

Claim. The action of on is ergodic.

Before proving the claim, let us start with a general fact.

Given an action of on , when is the restriction to ergodic? Answer is : iff the action of on is ergodic. Indeed, one can mod out by a proper action: the space of -orbits in , denoted by , is well defined, since the diagonal -action on is conjugated to the action on the -factor only, trivial on the factor. In fact, .

Applying this to the claim, ergodic on ergodic on ergodic on is implies by metric ergodicity of on .

This proves the claim.

3.8. Proof of superrigidity theorem

Again, fix an action of on a Lebesgue space which is amenable and metrically ergodic. The action of on is amenable and metrically ergodic. According to Theorem 14, the gate is nontrivial, i.e. .

Pick a generic -orbit in , identify it with . Get a map . The ergodic action of on yields a gate , the normalizer of in . Mod out by , get a -map . By ergodicity, we get a map .

3.9. Assume that

By simplicity of , . So the -action on has a nontrivial representation , whence a -representation . There exists a gate where . By ergodicity of the -action on , we get a -equivariant map (where , where is the normalizer of ) from to . Again by ergodicity, there is a -invariant point in . It also fixed by the Zariski-closure of , which is . Hence , is normal in . By simplicity, . So the above morphism to was to . The formula

defines an -invariant (hence constant) map . In other words, . This implies that . Composing with the projection , we get whose restriction to equals . So we are done under the assumption that . This was the easiest case.

3.10. From no on, assume that

Then , so . We have an -equivariant map (equivariant w.r.t. ).

Compose this map with , the diagonal. This is an AREA of to . By ergodicity of on , there is a gate where . Let us show that we are done if .

Assume that . Let us mod out the left -action. This mods out the -action in the gate, thus I get an -map from to a nontrivial quotient of . As before, we get an extension of to .

From now on, assume that . acts on . Since is equal ti its own normalizer in , the gate factors via a map , so extends as before.

4. Margulis superrigidity

Let be a real semisimple group of higher rank, let be an irreducible lattice. Margulis superrigidity states that every Zariski-dense unbounded homomorphism uniquely extends to .

The case when is a product has been treated. Next time, I will prove the case when is simple. I explain now that we are not too far from it.

Consider , . It is amenable, noncompact, and its centralizer is the upper diagonal . The -action on is amenable and metrically ergodic. acts on and we get a gate .

I would like to indicate a related geometric context without groups acting, that of -buildings, covered by Caprace-Lecureux. At infinity, such buildings have an exotic projective plane. It has a large group of projectivities (in fact a pseudogroup of maps from lines to lines). This yields a large group acting on the boundary of a tree. It plays the role of . It is not too hard to show that this group is linear iff the building is classical. It follows that is linear iff the building is classical.

4.1. Remarks on lattices in products

1. The superrigidity theorem holds for lattices in products of factors for any .
2. Nonarithmetic examples are known only for yet.
3. The examples are Kac-Moody groups acting on twin buildings and Burger-Mozes and Wise examples acting on products of trees.

4.2. A new category of representations

For the proof of Margulis superrigidity, we need to modify the concept of AREA. Up to now, we represented an action on Lebesgue space . Now we need to represent pairs of closed subgroups of a locally compact second countable group .

The objects in our category are now the following data:

• two closed subgroups and of ,
• an algebraic -group ,
• a -algebraic subgroup that commutes with ,
• a homomorphism and a continuous and Zariski-dense homomorphism ,
• a representation, i.e. a measurable map which is -equivariant.

Theorem 20 Let be a lattice. Assume that the action of on is ME. Then there exists an initial object (a gate), i.e. a -equivariant map

where and .

The proof follows similar lines as Theorem 11. Pick a minimal .

4.3. The nontriviality theorem

Theorem 21 Let be a lattice. Assume that the action of on is ME. Assume further that is amenable, is -simple and is unbounded, then the gate is not trivial.

Indeed, consider the action on . By amenability and ME, there exists a proper subgroup and a representation , under homomorphism . The gate will be a deeper object, but this suffices to prove that the gate is nontrivial.

4.4. Functoriality

Assume that is a noncompact simple -algebraic group. Let be a lattice. Recall that for every closed noncompact , the actions of on and of on are ME (this is Howe-Moore’s theorem). We now show that this gives right to extra invariants.

Fix , . Consider the category of -actions, i.e.

• the objects are closed, noncompact subgroups ,
• the morphisms are elements of acting on the right and conjugating the right action of into the right action of on .

Consider next the category of -spaces:

• the objects are ‘s acting on -space , commuting with ,
• the morphisms are –-morphisms of varieties .

The gate functor assigns to a -action of on an orbit with action of . To a morphism given by element , the gate associates a unique morphism of algebraic varieties .
In particular, we get a map . The group . On the other hand, . This homomorphism is a nontrivial datum: it gives extra invariance to the representation .

Corollary 22 If normalize each other. Then they have the same gate .

I really mean, the same map serves as a gate for both.

4.5. Proof of Margulis superrigidity

Recall our goal. Let be a real semisimple group of higher rank, let be an irreducible lattice. Margulis superrigidity states that every Zariski-dense unbounded homomorphism uniquely extends to .

Fact. There exist closed noncompact abelian subgroups generating such that each commutes with .

In fact, this is equivalent to higher rank.

Example. . The first three ‘s are -parameter subgroups of the Heisenberg group of upper unipotent matrices, the three next are -parameter subgroups of the opposite Heisenberg group of lower unipotent matrices.

Geometrically speaking, it means that one can move from any geodesic to any other by travelling within finitely many maximal flats.

Now we embark in the proof. By Corollary 22, all the ‘s have the same gate , equivariant with respect to . By ergodicity, the -invariant point is invariant under the Zariski closure of , i.e; b y , so , is normal in . Since is amenable, so .

So the gate is , equivariant with respect to .

defines a pull-back map from the algebra of polynomial functions on , , to -valued functions on , . It is injective, since its Zariski support is . This embeds as a -invariant subalgebra of for all , hence an -invariant subalgebra. This gives an -action of on extending the actions of the . Thus acts on on the right, when a homomorphism . The gate is -equivariant, this implies that extends to . End of proof.

4.6. Rank one

Let be a real semisimple group of rank one, let be an irreducible lattice. Let be a Zariski-dense unbounded homomorphism. What extra assumptions should we make to show that extends?

For , let , , , ,… denote the -gate.

Lemma 23 Let be a parabolic. If , then extends.

Indeed, consider , hence , .

Since and generate , extends as before.

Lemma 24 Assume that there exists a simple, noncompact subgroup and a -representation , equivariant with respect to , with . Then is not in the kernel of .

Indeed, is noncompact. By a finite number of up and down steps of taking normalizers we reach (two steps should be sufficient). All these intermediate normalizers have the same gate. Assume by contradiction that . Then the -gate equals the -gate, which thus factors by . This map is both and -invariant, but and generate . Thus , contradiction.

Note that the assumption of Lemma 24 never holds if and is nonarchimedean.

4.7. Relation to arithmeticity

Definition 25 Say that is compatible (with ) if for all proposer subgroups of , , where .

Let where . Le ne an orreducible lattice.

Fact. There exists a unique minimal number field such that is defined over and up to conjugation and finite index.

Theorem 26 (Margulis arthmeticity criterion) is arithmetic if and only iff its image is precompact in any place other than . I.e. for every embedding to a local field, either is precompact or and .

Supperrigidity implies arithmeticity. Indeed, take . Can assume that it is adjoint. For every embedding , the image of the composition is precompact unless . By an argument of Borel-Tits, one gets .

Observe that one need only a few targets to get arithmeticity.

Theorem 27 (Bader-Fisher-Miller-Stover) If there exists in infinitely many immersed maximal totally geodesic subspaces of dimension , then the assumption of Lemma 24 holds for all ‘s relevant to arithmeticity.

Corollary 28 is defined over the ring of integers of a number field.

Exercise. For , , the group is compatible. In particular, if there exist infinitely many immersed maximal totally geodesic subspaces of dimension , then is arithmetic. (Special care is needed for ).

5. Apafic Gregs

Talk given in Fanny Kassel’s seminar, on Oct. 11th, 2021, dedicated to Margulis, Perlman and Baldi.

Joint work with Alex Furman.

How to produce a random element of a group? In fact, a sequence. We will study linear representations of such random elements.

5.1. GREGs

Definition 29 Let be a probability space. Let be an ergodic invertible pmp transformation of . Let be a locally compact second countable group. Let be a map. The Greg is the data .

Examples.

Compact. Let , , an irrational rotation.

Random walk. Fix and a probability measure on . Let , shift, projection to the -th coordinate. The resulting Greg is a random walk.

Markov chain. Fix a graph with probability transitions (i.e. for each vertex , a probability measure on the set of edges emanating from ), set of paths in the graph, some Gibbs measure associate. Decorate edges with elements of . This defines a map .

Geodesic flow. Let be a compact negatively curved Riemannian manifold. Let denote the unit tangent bundle of . geodesic flow. Let . Fix a fundamental domain. For each time , there is a map which tells which group element drags back to the fundamental domain. One can decorate the picture with a flat bundle.

5.2. Associated constructions

Let be the minimal -algebra such that is measurable. Given , let . Assume that .

Let equipped with the future -algebra . Let equipped with the past -algebra (I view these as factors of ). Then maps , . It turns out that can be reconstructed from the pair , using the natural extension construction.

Definition 30 Let be a pmp map which is not invertible. One constructs another system, with an invertible transformation, by taking the inverse limit of

Example. , shift, then the natural extension is .

Theorem 31 If is ergodic on , then its natural extension on is metrically ergodic.

Here, metrically ergodic means that for every metric extension (i.e. a family of complete separable metric spaces on which a lift of acts isometrically, there exists an equivariant section.

Fact. The map defines a cocycle as follows. For , , and for ,

The cocycle identity

holds. Then a map is defined by

Whence an action of on .
The projections and , when modded out by , yield the factors and . If instead one mods out by , one gets the space of -ergodic components of , and factors and . I think of these as ideal futures and pasts.

Theorem 32 , and are amenable -spaces, the maps and are -metrically ergodic.

5.3. Asymptotic past and future independence condition

Definition 33 Say that the asymptotic past and future independence condition (apafic) is satisfied if the map is measure class preserving.

In other words, if you were born poor, you still may become rich. It holds for all examples above but the compact example.

If the Greg is Apafic, then the pair is called a Boundary pair. The spaces are amenable and the maps and are metrically ergodic.

Theorem 34 Let be a simple -algebraic group. Let be Zariski-dense. Then there exist opposite parabolic subgroups and measurable equivariant maps , and .
Moreover, if , is the minimal parabolic.

View the Cartan projection as a leftinvariant -valued metric on . Let be the composition of with and Cartan projection,

Subadditivity shows that converges to an interior point of the Weyl chamber . This is called “simplicity of the spectrum”.
I view apafic gregs as a useful generalization of random walks, and we see that the simplicity of the spectrum theorem holds for them.

Tholozan, Ledrappier: See Avila-Viana for the Markov chain version.

> Let be a simple -algebraic group. Let be Zariski-dense. Then there exist opposite parabolic subgroups and measurable equivariant maps , and .
Moreover, if , is the minimal parabolic.

View the Cartan projection as a leftinvariant -valued metric on . Let be the composition of with and Cartan projection,

Subadditivity shows that converges to an interior point of the Weyl chamber . This is called “simplicity of the spectrum”.

I view apafic gregs as a useful generalization of random walks, and we see that the simplicity of the spectrum theorem holds for them.

Tholozan, Ledrappier: See Avila-Viana for the Markov chain version.

Notes of Karim Adiprasito’s Hadamard Lectures 2021-04-19

Karim Adiprasito

I want to show how combinatorial tools can be used to prove results which previously came from algebraic geometry.

Today, I give an

1. Overview of applications

1.1. Matroids and characteristic polynomials

Example. The chromatic polynomial of a graph counts the number of colorings with colors. To compute it, one can use inclusion/exclusion, removing an edge and putting it back.

A matroid is an abstraction for the concept of linear independence in vectorspaces.

Example. Given a configuration of vectors in a vectorspace, let be the number of independent sets of size .

Often, the sequence of coefficients has a log concavity property:

(in the chromatic case, the signs are alternating, but this does not affect the inequality).

1.2. Face vectors in embeddings

A seemingly simple problem. Let be a simplicial complex. I want to embed it PL into a vectorspace . Subdivision is allowed. How many faces are needed? Let be the number of faces of dimension after subdivision.

Theorem 1 A necessary condition is

This is asymptotically tight.

Example: for an embedding of a graph in the plane, . For a -complex in , the numbers of triangles and edges must satisfy

Here is why it is easy for graphs. Adding edges, one can modify the embedding in order that every complementary region is a triangle. Then one uses Euler’s formula and . The point is that the modification changed only the left-hand side of the inequality.

The 2D case is much harder.

1.3. Triangulations of manifolds

Let be a closed manifold, fix a triangulation, the -vector counts simplices of each dimension starting from .

Theorem 2 The -vectors of simplicial spheres are realized by -vectors of simplicial polytopes, and are characterized by an algebraic condition.

To make a polytope into a simplicial polytope, one merely needs put its vertices in general position.

Later I will explain this algebraic characterization.

2. Basics

2.1. The algebra associated to a simplicial complex

Let be any infinite field. Let be a finite simplicial complex (by convention, the empty set is a face of ). Consider the polynomial ring with one indeterminate for each vertex of . Let be the ideal generated by all monomials which are not supported on . Let

Let

denote the Hilbert polynomial of . If has dimension ,

In order to remove a useless infinite dimensional part in , we pick a matrix whose rows are linear combinations of vertices. The quotient

is of Krull dimension if and only if the rank of the restriction of to is equal to . This holds generically.

2.2. Cohen-Macaulay

An interesting case. Assume is Cohen-Macaulay. I.e. if I define the length of to be the Krull dimension of , then is a regular system for , i.e. if multiplication by is injective on . If

then

In this case, can be computed as follows. Write a Pascal triangle with -numbers on the right-hand side, ones on the left-hand side and the skew-Pascal rule (every entry is the difference of both entries above it). Then the coefficients of appear on the bottom line. Conversely, one can recover -numbers from the coefficients of (the -vector).

Theorem 3 (Hochster) is Cohen-Macaulay if and only if for all nonempty simplices in , the link of in has homology concentrated in dimension .

MK: Gorenstein corresponds to links being homology spheres? Yes, see below.

Theorem 4 (Macaulay) (-vectors and) -vectors of Cohen-Macaulay simplicial complexes are the same as Hilbert polynomials of commutative graded algebras generated in degree .

They are sometimes called -vectors.

2.3. Poincaré duality

Now we go to MK’s spoiler.

Boundaries of simplicial polytopes are a source of simplicial spheres. Think of as a fan. Consider the algebra of cone-wise polynomial functions. If is rational, is isomorphic to the equivariant cohomology of the toric variety over . Each vertex of defines an element of , which is linear (equal to at ) on $latex {{\mathbb R}_+ v}&fg=000000$ and on all other vertex rays. These generate . Multiplying vertex functions gives simplex functions $latex {\chi_\sigma}&fg=000000$. Let us mod out by globally linear functions. Claim: the result is , where is given by vertex coordinates in ambient .

Theorem 5 (Poincaré duality. Probably due to Hochster) Let be the boundary of a homology sphere of dimension . Then the degree component of is -dimensional, and for all , the pairing

is perfect.

Here, a homology sphere means that is Cohen-Macaulay and the -homology of every link is concentrated in top dimension and has dimension .

I will explain where this comes from just below.

2.4. Shellability

Say a simplicial complex of pure dimension is shellable if or is a -simplex or there exists a -simplex in such that the simplicial complex (induced by other -facets) is shellable and is shellable of codimension .

For instance, two simplices touching at a vertex do not form a shellable complex, since has codimension . On the other hand, two boundaries of simplices touching at a vertex is shellable. In general, boundaries of simplicial polytopes are shellable. Proof by picture: start from a facet in and launch a rocket. You travel in the rocket and look back at the earth. Remove facets in the order in which they appear in view. Go to infinity and come back the opposite side.

Shellable implies Cohen-Macaulay over all fields. Indeed, it implies that is homotopically equivalent to a wedge of -spheres.

For shellable, let the facet removed in shelling step. Then the kernel of the (surjective) restriction map

is generated by , where is the minimal simplex of not in (everything here once modded out by ).

If , given an element , I first express it in shelling steps. The shelling turned around (in reverse order), is a shelling too. Therefore I can construct an element step by step that pairs nontrivially with .

3. Hard Lefschetz theorem

Today, I want to go to the hard Lefschetz theorem and its combinatorial applications, “beyond positivity”.

Let be the boundary of a -dimensional polytope . Remember the algebra is expressible in terms of cone-wise polynomials:

Let be a cone-wise linear function. Assume that is strictly convex (i.e. the domains of linearity are exactly the cones of the fan). Think of as defining an ample line-bundle.

OG: should one assume is rational? No, this is not necessary. It allows to interpret the algebra in terms of a toric orbifold, but we shall not use it.

Theorem 6 (Hard Lefschetz theorem) For all , multiplication with yields an isomorphism .

Remember that when is Cohen-Macaulay over , the -vector, given by dimensions of homogeneous components , is an M-vector (i.e. realizable as the dimensions in a polynomial ring). Define, for ,

Then Lefschetz theorem implies that the -vector is again an M-vector. This observation is due to Stanley. In fact, the converse is true: all M-vectors arise in this way (Billera-Lee).

3.1. Hodge-Riemann relations

Define a quadratic form on as follows.

The primitive part is the kernel of the multiplication by

Theorem 7 (Hodge-Riemann relations) The quadratic form is definite of sign on .

Let be two strictly convex elements of . The restriction of to the span of and cannot be definite, because it has a plus from degree 0 and a minus from degree . Therefore

This is Alexander-Fenchel’s inequality, that states that for convex sets in , the function

is log concave.

3.2. Application to Grunbaum’s conjecture

Question 1. We know that Poincaré duality holds in a broader generality. Does Stanley’s observation (“-theorem”) extend to simplicial -homology spheres?

Assuming this, I prove

Theorem 8 Let be a simplicial complex embedded PL into . The number of faces satisfies

Think of as a subcomplex of a triangulated -sphere (see Bing’s notes on the Geometric Topology of -manifolds, 1983). There is a restriction map : . Faces provide a generating system for (because is generated, as a vectorspace, by square-free elements), therefore

(Remember that the length of is the Krull dimension of , which is larger than the Krull dimension of . Later on we shall see a model for where this inequality will become obvious).

Relations come from codimension faces of -faces, therefore

Claim. . Indeed, by Poincaré duality and if Lefschetz property applies to , then a commutatiove square holds, and multiplication by is surjective

Therefore

This completes the proof of Theorem 8.

Our next goal is the following theorem.

Theorem 9 (Adiprasito 2018, Adiprasito-Papadakis-Petrotou 2021) Let be a -homology sphere (i.e. is Gorenstein*). For a generic choice of and a generic choice of , the Hard Lefschetz theorem holds for . Furthermore (2018 version), the restriction of quadratic form to any square-free monomial ideal is perfect.

Note that I do not claim any positivity, merely nondegeneracy. OA: sign main depend on the choice of .

Continuation of theorem (2021 version). In characteristic , there exists a field extension of such that for all nonzero .

4. The partition complex and Poincaré duality

Now we go to cohomological tools.

4.1. The works

How do I prove Poincaré duality? I show that the socle of has dimension . For this, I need two ingredients.

a) For all , the map to stars of vertices

is injective.

b) For every vertex , the multiplication by the indeterminate

is injective.

Let us prove b) first. I ignore modding out by . By Cohen-Macaulay property, the exact sequence

(where the central map is multiplication by ) stays exact after modding out, for all .

Let us prove a) now. The complex

with suitable choices of signs, is exact is positive homogeneity degrees (in degree , this is the \v Cech complex).

Consider the Koszul complex . The tensor product

is a bi-complex. It is exact in the Koszul direction, and exact in the in positive homogeneity degrees. Thus the diagonal complex is acyclic in positive homogeneity degrees.

If is Buchsbaum (i.e., it is of pure dimension and for every vertex, the star is Cohen-Macaulay), the kernel of the map is isomorphic to the cohomology

In general, if is a closed orientable simplicial manifold of dimension , is not Poincaré duality, but the quotient

is.

4.2. The cheats

Let me state the most general version of the Hard Lefschtez theorem we have now.

Let be a simplicial complex of dimension . Let be any -dimensional quotient of . The kernel is always . Then $latex {B^(\mu)}&fg=000000$ is a minimal quotient of $latex {A^(\Delta)}&fg=000000$ such that is isomorphic to .

Let be an element in which is dual to . Let be a simplex of cardinality . Let

be the corresponding monomial. Then

where is the minor of in the matrix .

Fix . For a generic choice of (and generic ), satisfies the Hard Lefschetz theorem.

5. Proof of the classical case of Hard Lefschetz Theorem

I first state McMullen’s version (“On simple polytopes”, following Fleming-Karu 2009.

Theorem 10 (McMullen) Let be the boundary of a simplicial -polytope. Let be a strictly convex cone-wise linear function.

HL:

is an isomorphism.

HR: is definite of sign o n primitive elements .

Next is the nonclassical version (due to Adiprasito 2018, Adiprasito-Papadakis-Petrotou 2021).

Theorem 11 Let be a triangulated -sphere. Thee exists and there exists such that

HL:

is an isomorphism.

HR: is nondegenerate on any squarefree monomial ideal.

OG: does this extend to other spaces than spheres?

Good spoiler! No. For , , consider the monomial ideal generated by . Then elements of this ideal are cone-wise linear functions that vanish on a half-plane, they must be restrictions of globally linear functions on the opposite halfplane, hence the square vanishes.

5.1. Classical proof

Today, I give the classical proof. It is a double induction.

5.2. The climb (raise in dimension)

Lemma 12 Let . Let be strictly convex. Assume that the theorem (HL and HR) holds in codimension . Then HL holds for .

Proof is by contradiction. Assume there exists in such that . For every vertex of , there is a restriction map to the link . The restriction of to this fan is still strictly convex. Thus the image of in the sum of links is nonzero. Since

We compute $latex \displaystyle (-1)^k Q^\Sigma(\alpha,\alpha)=(-1)^k \mathrm{deg}_\Sigma(\ell^{d-2k}\alpha^2)=(-1)^k \sum_i \mathrm{deg}_\Sigma(\ell_i x_i \alpha^2 \ell^{d-2k-1})&fg=000000$

$latex \displaystyle =(-1)^k \sum_i \ell_i\mathrm{deg}_{lk_v}(\alpha_{|lk_v}^2 \ell_{|lk_v}^{d-2k-1}).&fg=000000$ In the last term, the sign of the degree is provided . The sum is strictly positive. This proves that , and HL follows.

5.3. The hike (stay in the same dimension)

Next we prove HR. First assume that is the boundary of the -simplex. Then , and HR is proven by direct calculation.

Next we deform a general simplicial polytope to the simplex continuously. Start with the simplex, add vertices in general position in order to have as many as in and let them move. Assuming general position, for all but finitely many times , the obtained convex hull is simplicial. In each interval, HL is true hence HR is preserved. So we need only understand what happens when crossing special times.

Assume that . What we observe are Pachner moves: – flip of diagonals in a quadrilateral face, – upraisal of a vertex in the middle of a face. In the first case, there is a common refinement (blow up by adding a vertex at the intersection of diagonals). I denote by the cardinality of the minimal simplex introduced (). I call the side where this minimal cardinality is smallest. I denote by the flip locus after the flip (it is the star of the minimal simplex introduced). Then

is the part which is unaffected by the flip. My goal is to construct a map .

I introduce relative algebras: in the can of boundaries of polytopes, is the ring of cone-wise polynomials on that vanish on . In the general case,

where is the ideal generated by monomials whose support is not in .

The following sequences are exact:

The generator for is in homogeneity degree . The generator for is in homogeneity degree . Thus, for , the maps

are isomorphisms. For , the maps

are isomorphisms. Therefore, grosso modo,

where truncated after . This decomposition is orthogonal with respect to the Poincaré pairing. Indeed, if has degree and has degree , then which injects into , therefore . If instead is the generator of of degree , i.e. of minimal face introduced, then is a product of concave cone-wise linear functions, so the quadratic form is

its sign is .

This prove the HR relations.

5.4. Beyond polytopes (but still convex)

I explain the application to matroids. A matroid is a pair where is a finite set and (ordered by inclusion) is a set of subsets of satisfying

• , .
• .
• If , the set partitions .

The fan associated to in , , is defined as follows. Let $latex {(e_i)_{i\in E}}&fg=000000$ be a generating set in such that . For every subset , set $latex {e_F=\sum_{i\in F} e_i}&fg=000000$. To a flag of , we associate the cone on the vectors . Then the fan is the collection of all theses cones.

Example. The set of singletons in a set . The fan consists of only one cone, the whole plane.

Example. The full power set . The fan consists of 6 cones in the plane.

Theorem 13 (Adiprasito, Huh, Katz) Let be a matroid. The algebra

satisfies Poincaré duality (the socle degree is the longest chain in ). Hard Lefschetz. Hodge-Riemann.

Next time, I will explain the non classical Lefschetz theorem. Before, I will explain a bit of the context of matroids.

OG: worries about the notion of ampleness.

JK: does the ring you define correspond to the Chow ring in algebraic geometry? Yes.

JK: does the proof follow the same inductive scheme as McMullen’s ? Yes.

MK: do secondary polytopes arise in matroids? yes.

6. Matroids, continued

Today, I quickly finish with matroids and then proceed to Lefschetz without positivity.

Theorem 14 (Adiprasito-Huh-Katz) Let be a matroid, let denote its Bergman fan. Then the algebra satisfies

• Poincaré duality (the fundamental class belongs to the degree corresponding to the longest chain).
• If is ample, then Hard Lefschetz and Hodge-Riemann hold.

6.1. Characteristic polynomial of a matroid

Given a vector , a matroid can be reduced in two ways:

1. By removing , get .
2. By projecting mod (and removing ), get .

Recursively, let us define the polynomial (in indeterminate )

starting from the following chromatic polynomials:

(this gives rather the characteristic polynomial divided by ).

Question. Can one give a formula for ?

Empirically, one sees that coefficients alternate in signs and their absolute values are unimodal, i.e. increase and then decrease.

From the above theorem, it follows that

One can access to from certain intersection numbers. Let denote the degree of the fundamental class. The relevant intersection numbers are

where

where is a subset of and , and

Then all (resp. all ) are the same (independent of ). This is because one mods out by global linear functions. Denote them by and . These classes are nef. They can be approximated by ample classes.

I will not prove that is a coefficient; merely explain its combinatorial content.

The Hodge-Riemann relations imply the log-concavity. Indeed,

can be interpreted as the determinant of a matrix which is indefinite, so the determinant is negative.

The proof, in the case with positivity (ample classes) follows the same lines as McMullen’s argument for polytopes: induction.

7. Hard Lefschetz without positivity

I will focus on the following special case. is an arbitrary infinite field.

Theorem 15 Let be a triangulated -sphere (meaning a homology manifold, which is a homology sphere). Then for a generic Artinian reduction and a generic element ,

• the Hard Lefschetz theorem holds
• Hall-Lamac relations hold, i.e. does not degenerate on squarefree monomial ideals.

Assume that . The first nontrivial isomorphism is multiplication by : .

Let be a vertex. The multiplication with is restriction to the star of . Its kernel is the relative space . Its image is . Let be an other vertex. One expects that the kernel of multiplication with a generic linear combination of and be as small as possible, i.e. . Dually, the image of this combination should be . Now is expressible in terms of the homology of the intersection of two stars, i.e a link, which vanishes.

The following classical fact supports this.

Lemma 16 (Kronecker) Let and be two vectorspaces over . Let be linear maps such that and intersect trivially. Then the kernel of a generic linear combination of and is .

Proof. It suffices to show that and .

Here, and are dual to each other, so one can speak of orthogonal complements. Also, is the orthogonal complement of .

So now we see that constructing an isomorphism is related to nondegeneracy of the pairing on certain subspaces.

7.1. Biaised pairing theory

Again, we want to prove inductively the “transversal primes property” for all subsets of the vertex set of (by adding vertices one by one): for a generic linear combination of ‘s, , the kernel of the multiplication is the intersection of kernels. If we can do it for all vertices, we shall be done for Hard Lefschetz.

7.2. The genericity of is needed

Let me construct a counterexample. The simplest sphere is the tetrahedron, boundary of the simplex on . has Krull dimension , so consists of linear forms. Let us pick generic vectors for the two first and for the third. This is not a linear system of parameters, so we must modify it. Subdivide each facet, this produces more vertices , hence we continue the linear forms with generic vectors. We must check that minor corresponding to do not vanish. This now works. No choice can satisfy Hard Lefschetz.

7.3. Biased pairing property

Say that and ideal of satisfies the biased pairing property (bpp) in degree if

is nondegenerate in the first factor.

Say that and satisfies the biased pairing property in degree if all its squarefree monomials ideals do.

We use a descent lemma.

Lemma 17 Let be a -sphere. Let . Then satisfies the biased pairing property in degree if all links of vertices do.

So we only need to consider the middle case, in degree , when the sphere has dimension .

From now on, this is the case. I will explain a special case, when the ideal is the nonface ideal of a subcomplex , i.e. the kernel of

where is a codimension sphere in . Denote by and the two hemispheres determined by . Then

is an orthogonal splitting. So we merely need prove the bpp for .

Observation 1.

since

Observation 2.

Lemma 18 Say we want to prove bpp for an ideal of in degree . This is equivalent to an injection

in degree .

Let us combine these two observations. When , the annihilator of is , hence

So we want an injection

Write . Before Artinian reduction, the map

is an injection. Since all terms in the sequence are Cohen-Macaulay, a generic choice of is a regular system of parameters for , and the sequence remains exact when modding out by .

Since is a -sphere, by Poincaré duality, multiplication by ,

is injective, and thus bpp for in degree is equivalent to Lefschetz property for

Since is a lower dimension sphere, this allows for induction.

8. Hard Lefschetz in codimension implies bpp

Last time, I explained how to iteratively construct a Lefschetz element using Kronecker’s Lemma. Specifically, bpp (the nondegeneracy of the Poincare pairing at ideals) in codimension inmplies the existence of Lefschetz elements.

To close the loop, I must show that Hard Lefschetz in codimension implies bpp.

Later on, I will give another proof, based on transcendental arguments (no induction).

Last time, we dealt with ideals of the form where was a codimension sphere. Today, I cover the case where is a codimension manifold which is -acyclic over . is a trivial case.

As before,

where separates into two pieces and .

Remember that the partition complex yields a map

Theorem 19 satisfies bpp if and only if the composition

is an isomorphism.

This is mainly diagram chasing. Under the assumption of the theorem,

is an isomorphism, where

This is exactly what we want.

8.1. How to prove that satisfies bpp in general?

Assume again that has dimension and has dimension . The idea is to construct a hypersurface such that contains and is isomorphic to .

Assume that is a PL sphere.

Observation. Subdivision of does not affect the fact that has bpp (we shall only use stellar subdivisions). Indeed, injects into , the subdivision, and there is an orthogonal decomposition

Since consists of all monomials not in , the same orthogonal decomposition holds for

so the required nondegeneracy holds after subdivision if and only if in holds before.

Since can be embedded in the boundary of its regular neighborhood (radial projection), there exists a refinement of , not affecting , such that

This provides us with a sujective map

is not exactly the hypersurface we need. Every element of the kernel of that map can be killed by introducing holes. This produces a hypersurface with boundary . It is orientable. Double it and compactify it. By this, I mean I use scissors to open along . Then becomes , a closed hypersurface, in a new which is still a homology sphere. By diagram chasing, one shows that satisfies bpp if and only if does,

8.2. Summary of what we have done

1. Kronecker’s Lemma allows to construct Lefschetz elements in provided we can prove nondegeneracy of pairing

holds.

2. Nondegeneracy of pairing at is implied by Hard Lefschetz for hypersurfaces. Using a hypersurface containing , with covers the general case.
   
   Each step wins a dimension.
   
   There is an issue: the above kernel, intersection of two monomial ideals, is not monomial in general. The above kernel is the orthogonal complement to a monomial ideal, provided

and

are submanifolds. This fact follows from shellability, but it is too strong an assumption.

Recall that we want Hard Lefschetz for a dimensional sphere. Order vertices arbitrarily. Embed into a -manifold . Then Hard Lefschetz holds for in degree if and only if satisfies bpp in degree , i.e. if and only if satisfies bpp. We manage to find a refinement of such that there exists a hypersurface with boundary such that is unchanged, and…

9. Transcendentality

Let be a simplicial -cycle of dimension . I mean a pair where is a simplicial complex and is a -cycle in it, hence represents a class in .

On defines

The top-degree chain can also be viewed as a cochain . It defines a pairing of degree to . I view as a quotient of , hence of . Let be the smallest quotient of such that . Then

is a Poincare duality algebra, with fundamental class in degree . Does satisfy Hard Lefschetz?

Consider the field extension of generated by extra algebraically independent elements, the rows of and .

Theorem 20 Let be a -cycle over . Then, over ,

1. satisfies Lefschetz property.
2. satisfies the Hall-Lamac relations.
3. If , then for all nonzero elements .

Open question: what happens in other characteristics?

9.1. Corollaries

Observe that if is a sphere, then

In general, if is an orientable, closed manifold, we define

Corollary 21 If is an orientable pseudomanifold, then satisfies Hard Lefschetz.

We are now very far from any geometric interpretation.

9.2. The degree function

Define as follows. For a facet of of dimension , set

where denotes the minor in the matrix defined by .

AB: Morozov’s Lemma.

10. Transcendentality (continued)

Recall that is the field of rational functions in and the rows of . Today, we want to prove the following

Theorem 22 Let be a -cycle over . Then, over ,

1. satisfies the Hard Lefschetz property.
2. satisfies the Hall-Lamac relations.
3. If , then for all nonzero elements .

11. Basic reductions and lemmas

11.1. Reduction to the middle dimension

Let be a -cycle. Let be a graded ideal in , . Then satisfies Hall-Lamac relations with respect to in degree if

does not degenerate on . We want to establish this by only considering .

The suspension of is the free join of with a -cycle, i.e. where and are collapsed to distinct points and . Then is a -cycle. Let be a linear system of parameters (for an Artinian reduction) for . Projection along yields a linear system of parameters for .

Theorem 23 Assume that . Let be any ideal in . The following are equivalent:

1. The Hall-Lamac relations for in in degree and with respect to .
2. The Hall-Lamac relations for in in degree and with respect to the coordinate .

Let be a subcomplex. If

where is the quotient of obtained like is obtained from .

Observe that

Let us prove Theorem 23. Consider the element of . Then there is a commuting square between multiplication by at the level of and multiplication by at the level of the suspension. Restricting to proves the theorem.

11.2. Squarefree monomial generation

Let be a simplicial complex. Then, for any linear system of parameters , is generated by squarefree monomials in each graded component.

I explain this for a generic choice of . I first construct a larger complex where it is true. Let be the -skeleton of the simplex on vertices. Let be a linear system of parameters for . Since is shellable, we know how to give a basis for when we attach a new face : we add a generator where is the smallest face of which is not contained in the complement of . Now generation by squarefree monomials for implies the same result for .

11.3. Continuation of the proof

Recall: if is a facet of , then

where denotes the minor in the matrix defined by .

Lemma 24 (Lee) Let be a simplex of cardinality , then

I want to evaluate for . Let and be disjoint simplices of cardinality in . Let me view them as differential operators. Let denote the element used to define the degree. Then is

i.e. a derivative after performing a column operation on .

Lemma 25

In particular, in characteristic ,

Thanks to Lee’s Lemma, if is any vertex not in , then is independent of .

11.4. Let me first treat characteristic

Let . We must show that . By Poincare duality, because is generated by squarefree monomials, there exists of cardinality such that . The ideal of of in degree is isomorphic to , and is again a cycle. is generated by , of cardinality . Therefore

Compute, using the lemma,

since all other terms

vanish. Hence

11.5. General characteristic

Let .

Definition 26 Say is compatible with if

1. The support of intersects is a single face and .
2. If is any other face in , then is a simplex , face of , or empty, and is independent of one of the variables of .

Let us extend again the ground field into using a second copy of . Let us associate to an element obtained by replacing with .

Lemma 27 For compatible with ,

If one substitues , one gets

By Poincare duality, we may assume that . If is in a monomial ideal of , then so are and .

There remains to show how to find a compatible element. For us, the ideal is for some subcomplex of , has dimension . Consider the pairing on , with values in . Let be represented by squarefree monomials. We know that there exists such that if , hence .

Intuitively, has dimension . We can assume that the class of in is represented by an element such that the support of intersect in a single face . Without loss of generality, we can assume that the coefficient is . Pick a face is the support of . I can assume that . Furthermore, is generated by a single element in top degree. This combines into compatibility.

11.6. Application

Definition 28 Say a simplicial complex is -Cohen-Macaulay if for every vertex , is Cohen-Macaulay.

Example. Triangulations of disks are not Cohen-Macaulay, but -Cohen-Macaulay.

Algebraically, -Cohen-Macaulay complexes are level, that is, for all , there exists such that (see Stanley’s book).

Let be a quotient of . Let us define

The level property implies that

Hard Lefschetz implies that multiplication by is an injection on . Thus we get

Corollary 29 (of Hard Lefschetz for cycles) The dimensions of are increasing while .

Beware that Poincare duality need not hold, so this does not imply unimodality of these dimensions (in fact, counterexamples exist). Nevertheless, such a corollary is valuable, since the class of -Cohen-Macaulay complexes is much wider that homology spheres.

Notes of Mikolaj Fraczyk’s YGGT IX lecture, february 27th, 2020

Growth of mod- homology of higher rank lattices and mapping class groups

1. Homology growth

Let be a finitely generated group, and a sequence of finite index subgroups. Let be some field. I am interested in the growth of Betti numbers. When

exists, what is it?

Note that if is of class , i.e. acts freely on a -connected complex , grows at most like the number of cells of , i.e. linearly with the index. Also, the Euler characteristic is proportional to the index, therefore linear growth is to be expected.

The answer is known in characteristic zero.

Theorem 1 (Lück 1994) If is of class , and are nested normal subgroups, then the limit exists and for all ,

is equal to an -Betti number.

What about other fields as coefficients?

1.1. Locally symmetric spaces

I especially like the following examples. If is a semisimple Lie group and a maximal compact subgroup, then the symmetric space has nonpositively curved left-invariant metrics, hence is contractible. Lattices exist, thery need not be cocompact. Nevertheless, they have finite covolume .

2. Benjamini-Schramm convergence

Lück restricted to normal subgroups, this restriction can be weakened. The replacement is a geometric condition: Benjamini-Schramm convergence. This means that for every , the relative volume of the -thin part tends to zero. -thin means the locus of points where injectivity radius is .

Theorem 2 (Abert, Bergeron, Biringer, Gelander) If the sequence converges to in Benjamini-Schramm’s sense, then

Again, this is over the reals, generalization to other fields is unclear.

Over the reals, one uses harmonic representatives, and spectral properties of the Laplacian. This is not available for finite fields.

3. Results

Theorem 3 (Fraczyk 2018) If is a higher rank symmetric space and converges to in Benjamini-Schramm’s sense, then

Ongoing work: growth of torsion.

Theorem 4 (Abert-Bergeron-Fraczyk-Gaboriau) For congruence subgroups of , and all ,

4. Sketch of proof of Theorem 4

One uses a less naive simplicial complex on which acts. The requirements are

1. is -connected,
2. .

Lemma 5 Assume that acts on a complex satisfying

• (A1) is finite.
• (A2) is -connected.
• (A3) Each simplex stabilizer is of class and contains a normal subgroup isomorphic to .

Then we can build the needed spaces for every normal chain .

Indeed, let . Using standard homotopy theory (see Geoghegan’s book, the keyword in stack of CW complexes), one glues together spaces into a space fibering over with fibers .

One counts cells,

Claim. The exist classifying spaces such that

This implies that and proves a weak form of Theorem 4. To win an exponent, we need control on norms of differentials.

Proof of the claim. We take for the Solomon-Tits complex, whose cells are index by -parabolics, and cells are incidents when parabolics are contained in each other. Vertices are maximal parabolics et top dimensional celles correspond to Borel subgroups. This complex is homotopic to a wedge of -spheres. The cell stabilizers are intersections of with parabolic subgroups (stabilizers of flags in case of ).

5. Proof of Theorem 3

Take . We want to estimate , assuming that injectivity radius tends to infinity.

We show that every class is represented by a cycle of length which is . Start with any representative, typically of length . If two pieces of the cycle are nearby, I can remove these pieces. Mod , this is possible. Then realize by geodesic. Due to Euclidean character, it can be wobbled inside maximal flats without increasing length too much. This puts again pieces close to each other. Repeating, one reduced length substantially.

5.1. Generalization

Our methods applies to Mapping Class Groups, with replaced with the curve complex.

Short presentations, YGGT IX, february 25th, 2020

5 minutes presentations

1. Anschel Schaffer-Cohen

I am interested in big mapping class groups. Let such a group act continuously on a graph , with unbounded orbits. Assume the graph is hyperbolic, plus some kind of other property. Then is quasiisometric to . This should lead to examples of hyperbolic mapping class groups.

2. Vladimir Vankov

I study cube complexes with groups acting. If one knows some finite quotient of , can one say something on . To each edge, there is a matrix associated matrix, containing the quotient. Take determinants, yielding characters. This has led sufficient conditions for cube complexes to be special.

An interesting situation is provided by the short exact sequence where a RAAG appears like an extension of by a Bestvina-Brady group.

3. Josh Faber

Let be finitely generated subgroups of a free group . The Hanna Neumann conjecture states that

where

Joel Friedman’s solution, using sheaves, suggests a new conjecture:

I proved a slightly stronger result, with a finer constant. I am also interested in generalizations to cube complexes.

4. Sam Hughes

I am interested in irreducible lattices in a product of groups which are either Lie or tree automorphism groups. Example: .

There exist nonuniform examples in the Lie group case. They are linear, just infinite. In the tree case, there exist irreducible lattices, sometimes simple, sometimes just infinite. I provide examples illustrating these properties in mixed products.

5. Nick Bell

Mirzakhani 2016: Fix a closed geodesic in a surface . The number of curves in of length is , where are known.

I have shown that this works for arcs as well.

6. Sam Shepherd

Leighton’s theorem states that if two finite graphs have a common universal cover, then they have a common finite cover.

Question: does Leighton’s theorem hold for other spaces?

I proved that the answer is yes in certain cases: graphs with fins, certain -complexes. I showed that answer is no for NPC cube complexes, but the question remains open for special NPC cube complexes.

Gardan-Woodhouse did it independently.

7. Lucas De Souza

I work on group compactifications. Gerassimov’s Attractor-Sum Theorem states that if acts properly and cocompactly on and has a convergence action on , a convergence action of can be defined on the join of and .

I introduce a property of a compactification of , called perspectivity, that garantees that such sums go through…

8. Shivam Arora

I study subgroups of hyperbolic groups. When are they hyperbolic? Note that hyperbolicity implies finite presentation. The converse fails (example by Bestvina-Brady).

Gersten proved that if is hyperbolic with cohomological dimension over , then every finitely presented subgroup is hyperbolic.

I have been able to replace with coefficients.

9. Sahana H Balasubramanya

I study the class of acylindrically hyperbolic groups. It contains relatively hyperbolic groups, most RAAGs, s, ,…

Acylindrically hyperbolic groups come with a generalization of peripheral groups, hyperbolically embedded subgroups. Are they QI invariant? Seems hard.

I prove that if is normal of finite index and AH-accessible, then is also AH-accessible (hence acylindrically hyperbolic). AH-accessible means that admits a largest acylindrical action on a hyperbolic space.

10. Feng Zhu

A really inefficient way of proving that a group is hyperbolic.

Let be finitely generated and torsion free. Say that a reprrsentation is -Anosov if for every element ,

Here, are singular values.

Kapovich-Leeb-Porti, BPS proved that if admits a -Anosov representation, then is hyperbolic.

The point is that is shows that simple Lie groups contain plenty of nonfree hyperbolic subgroups.

I proved a relative version of this.

11. David Sheard

Flag manifolds contain Schubert varieties, usually not smooth, sometimes rationnally smooth, this has led me to Coxeter groups. Indeed, there is a Weyl group associated with a flag variety. Rationnal smoothness is related with a Poincaré polynomial associated to .

There is an order on a Weyl group , the corresponding Poincaré polynomial is

Rationnal smoothness is equivalent to being palindromic. There is a criterion for that.

12. Nora Szoke

I would like to tell you what sofic groups are. An -approximation of a Cayley graph is a finite graph such that for most vertices of , the -ball at that vertex is isomorphic to the -ball in .

Say is sofic if and , there exists an -approximation. Obviously, residually finite groups are sofic. Amenable groups are sofic, because Folner sets do the job. The class of sofic groups is stable under direct products, subgroups, limits, free products. We do not know wether there exist nonsofic groups.

13. Dounnu Sasaki

I want to introduce the notion of subset current, due to Bowditch. Let be a hyperbolic group. Subset currents form a completion of the semigroup of positive linear combinations of quasiconvex subgroups. Reduced rank gives rise to a morphism to .

I showed that the inequality of the Strengthened Hanna Neumann’s conjecture for subgroups of a free group,

extends to subset currents. I think this is a useful tool.

14. Jonathan Fruchter

I want to study the first order theory of acylindrically hyperbolic groups. First order theory is the collection of sentences with only quantifyers, constants, variables and monomials.

Lubotzky proved that a finitely generated profinite group is uniquely determined by its first order theory. Free groups are not (Sela), they all have the same first order theory. Say a group has property (*) if .

With Simon André, we proved that if is acylindrically hyperbolic, then . Such groups also slightly a slightly weakened form of property (*).

15. Hannah Hoganson

According to Gromov, is nonamenable admits a Ponzi scheme. Thompson’s group acts on a rooted binary tree, with two generators. I have a good understanding of its Cayley graph. I have tried to construct a Ponzi scheme, without success.

Instead, I studied metrics on Teichmüller space. They turn out to be incomplete. The completion includes pinched curves, forming a stratification by lower genus Teichmüller spaces.

16. Xenia Flamm

I begin studying higher Teichmüller theory. This deals with homomorphisms of a surface group into a Lie group (up to conjugacy). A higher Teichmüller is a union of connected components of this space, consists of discrete and faithful homomorphisms. This exists for , , , for instance.

I want to understand compactifications of these higher Teichmüller spaces. A possible approach uses real spectra, following Brumfiel.

17. George Domat

Projection complexes, constructed by Bestvina-Bromberg-Fujiwara, deal with the projections to one of them of the lifts in the universal cover of a geodesic. These are quasitrees on which the fundamental group acts. This gives rise to quasimorphisms. I use this to show that the abelianizations of certain big mapping class groups contain infinite direct sums of copies of .

18. Yuri Santos Rego

To a conjugacy class in the free group, I associate a combinatorial object I call a “circle”. Artin did the same with braids and produced knots. Sapir and Guba did the same for diagram groups, Thompson groups is one of them. Inside the big MCG of the complement of a Cantor set, there is a “braided Thompson group” to which a similar procedure can be applied, the objects associated to conjugacy classes are Haken manifolds.

19. Thomas Ng

I like to decide wether two elements in a finitely presented group generate a free goup or not. Hard. Replace them by some high powers. Still hard. If group has some nonpositive curvature, there is some hope. Free semigroup would do.

With Radhika Gupta ans Junkiewicz, we prove that if acts on a -dimensional cube complex without global fixed point, for any generating set, either is virtually abelian or there exist distinct elements of length that generate a free subgroup. This applies to Higman’s group (which acts of a cube complex with stabilizers Baumslag-Solitar groups.

20. Macarena Arenas

In hyperbolic groups, two loops which bound an annulus bound an annulus with area linear in their lengths. With Wise, we generalize this to any number of loops, first in genus and then in higher genus.

21. Laurent Hayez

There are two distinct 6 vertex 7 edges graphs which have the same spectrum. I am interested in spectral properties of infinite graphs, and especially Cayley graphs. For those, there is a spectral measure .

Question. Does determine the group up to isomorphism?

22. Nima Hoda

Bisimplices are a trick to add higher dimensional cells to a bipartite graph, for instance to make it contractible. Cubes just won’t do (for instance, one cannot glue a cube to ).

Are there bipartite analogues to simplices? Answer: inductively define cells filling complete bipartite graphs. Use discrete Morse theory

23. Sami Douba

In a group , let be elements such that commutes with the others. Then in any finite dimensional unitary representation of , has finite order.

This implies for instance that Heisenberg group, fundamental groups of unit tangent bundles of surfaces, most mapping class groups cannot embed in a unitary group.

24. Matthieu Joseph

Let me share with you what I call the Infinite Noodle Lemma. I used it to build orbit equivalences between groups.

Theorem 1 Let a fixedpoint free involution on the set . I draw an arc in the upper half plane between and for every . Assume that the arcs do not intersect. Reflect and translate the picture by . Then one gets a curve.

25. Matthew Conder

Question. Given matrices , when is the subgroup they generate free?

For instance, there is no complete answer for

The ping-pong lemma gives a sufficient condition. If you ask simultaneously that be also discrete, i becomes easier: there is an algorithm that decides it (Eick-Kirschner-Leedham-Green).

I have solved the corresponding problem in , where it is easier, because ping-pong is easier there.

26. Holli Chopra

I plan to work on Cannon’s conjecture: if is a Gromov-hyperbolic group whose boundary is homeomorphic to a -sphere, does act geometrically on hyperbolic -sphere?

My approach is to modify Perelman’s methods.

27. Michal Buran

Let me tell you of nightmare towns, with randomly chosen railway system.

Two random permutations of generate with high probability, even if a number of constraints, encoded in a graph with vertices, are imposed.

28. Francisco Nicolas

Kähler groups are fundamental groups of compact Kähler manifolds. These are complex manifolds equipped with hermitian metrics whose imaginary part os a closed -form. Example: complex tori, Riemann surfaces, complex projective spaces and their complex submanifolds…

Which finitely presented groups are Kähler? is not, since the first Betti number of a Kähler manifold is always even (Hodge theory).

I plan to study finiteness properties of Kähler groups, by constructing examples of Kähler manifolds as level sets of functions, in the spirit of Bestvina-Brady.

Notes of Radhika Gupta’s YGGT IX lecture, february 25th, 2020

Non-uniquely ergodic arational trees in the boundary of Outer space

With M. Bestvina and J. Tao.

1. Surface case

1.1. Laminations

Let be a hyperbolic surface. A geodesic lamination is filling if the complement is made of discs or once-punctured discs. It is minimal if every leaf is dense in it.

Question. For measured laminations, does filling, minimal imply uniquely ergodic?

Minimal, filling laminations occur as early as 1969 in Veech’s work on interval exchange transformations.

Nonuniquely ergodic filling laminations appear in Gabai’s work.

Leminger-Lenzhen-Rafi have constructed nonuniquely ergodic minimal filling laminations.

1.2. Trees

The associated tree is the set of leaves in the universal cover, it inherits a metric from the transverse measure. If the lamination is nonuniquely ergodic, the tree supports two projectively distinct metrics: it is non ergometric.

2. Free group

Let denote the free group on generators. We collect all the simplicial actions of on trees which are free and minimal, up to homothety. Collapsing edges equivariantly allows to connect combinatorially distinct actions, it provides a topology, this is Outerspace . The outer automorphism group acts on it. We think that Outerspace plays for free groups the role played by Teichmüller space for surface groups.

has a boundary, consisting of isometric actions on real trees.

The analogue of the curve complex, a hyperbolic space with ideal boundary the space of minimal filling laminations, is the free factor complex, with boundary the space of arational trees.

Question. Do there exist arational trees which support non uniquely ergodic measures?

Theorem 1 (Bestvina-Gupta-Tao) There exist non-uniquely ergometric non-geometric arational trees.

2.1. Main construction

We start with two automorphisms of , which is fully irreducible on the first generators and identity on the others, and which cyclically permutes all generators

Set . One picks a sequence and let act successively . One orbit in is a zig-zag whose closure in compactified Outerspace contains an arc. On the other hand, the corresponding orbit in the free factor complex converges to some arational tree. This produces two different metrics on that tree.

Notes of Yair Hartman’s YGGT IX lecture, february 25th, 2020

Which groups have bounded harmonic functions?

Joint with Frisch, Tamuz and Vahidi-Ferdowsi.

Familiar groups act on boundaries

1. Dirichlet’s problem

Impose the temperature along the boundary of a domain , wait until equilibrium. What is the temperature inside? Temperature is a bounded function , harmonic in the interior. In case of a round disk, the solution is provided by Poisson’s transform. For general domains, it can be obtained as follows. Given inside, launch a random walk and record where it hits the boundary. This produces a probability measure on the boundary. The temperature satisfies

This provides a bijection between functions on and bounded harmonic functions on .

Note that all define the same measure class, so is unambiguous.

Liouville’s theorem states that all bounded harmonic functions on the plane are constant. I think of it in these terms: the boundary of the plane has only one point.

2. Measured groups

In order to define harmonicity, one merely needs a notion of average over neighbours.

Definition 1 A measured group is a countable discrete group equipped with a symmetric probability measure whose support contains a generating set. Symmetric means that .

A function is -harmonic if for all ,

denotes the space of bounded harmonic functions.

A notion of boundary for a measured group has been developped since the 1960’s.

Theorem 2 (Furstenberg) Given , there exists a standard probability -space such that

This is called the (Furstenberg-)Poisson boundary of .

Note that has no topology on it, only a family of measures.

The random walk interpretation is still valid. Given , pick independently elements according to and consider the sequence . Morally, is the set of trajectories and the hitting measure. This is litterally true when is a free group with the uniform measure on generators and their inverses.

Definition 3 Say is Liouville if is one point and has only constant functions in it.

Example. When is with mass on and , harmonic functions are arithmetic progressions. Only constant arithmetic progressions can be bounded.

Poisson boundaries behave well under taking quotients.

3. Which groups are Liouville

Question. Determine the classes of groups

Theorem 4 (Furstenberg, Rosenblatt, Kaimanovich-Vershik) is the set of nonamenable groups.

Theorem 5 (Blackwell, Choquet-Deny, Dynkin-Malyotov, Margulis) Virtually nilpotent groups belong to .

Theorem 6 (Frisch, Hartman,Tamuz,Vahidi-Ferdowsi) Among finitely generated groups, is the class of virtually nilpotent groups.

3.1. Proof

The key concept in our proof is

Definition 7 is ICC if all nonidentity conjugacy classes are infinite.

Starting from a finitely generated group, kill all FC central elements. In the quotient, do it again. Do it again and again (transfinite induction). Ultimately, one gets an ICC quotient. Since Poisson boundary behaves well under taking quotient, the problem is reduced to the ICC case.

Proposition 8 If is finitely generated and ICC, it admits a probability measure with nontrivial Poisson boundary.

The ICC property is exploited as follows.

Lemma 9 If is ICC, for every finite subset , there exists such that .

It means that starting with a ball, applying a suitable right translation blows it out into a sparse set of points.

We choose a probability measure concentrated at a sequence of such snipers leaving to infinity very fast. Two different trajectories will frequently encounter this blowing out phenomenon and stay far apart. They must exit by different points, showing that is more than one point.

The measures we construct are not finitely supported. Both variants of the above problems where one restricts to finitely supported measures are open.