## Notes of Richard Schwartz’ sixth Cambridge lecture 26-05-2017

PETs, pseudogroup actions, and renormalisation

Started by group theorist B.H. Neumann in 1959. Outer billiard around a convex polygon composes 180 degrees rotation through vertices.

Theorem 1 For a kite (a quadrilateral with one axial symmetry), there exist unbounded orbits iff the group generated by these rotations is indiscrete.

1. Square turning maps

Fix coordinates in the plane. The square turning map ${G_s}$ consists in tiling the plane by sidelength ${s}$ squares. Then turn each square by 90 degrees around its center. ${G_s}$ is discontinuous.

What does the group generated by ${G_1}$ and ${G_s}$ look like?

Especially interesting is the map ${G_1 G_s G_1 G_s}$, made of partial isometries whose linear parts are identity, i.e. translations. This is an example of a polygon exchange transformation (PET). Here, the polygons (called periodic islands) are rectangles.

For rational ${s}$, the pattern of rectangles is reminiscent of the continued fraction expansion of ${s}$. The pattern has flaws. The orbit gets longer, and fractal-looking, when the expansion gets longer. I am unable to explain this completely.

2. PETs

Interval exchange transformations arose as first return maps of measured foliations or rational billiards. A polygon exchange transformation is the same with intervals replaced by convex polygons.

2.1. Renormalization

The first return map on a polygon is often again a PET (sometimes, there are infinitely many pieces).

2.2. Compactification

This is a locally affine equivariant map of an unbounded PET into a bounded PET (i.e. a PET on a torus).

Example. An invariant line in a PET on a torus.

2.3. Construction of PETs

Given two lattices which share a common fundamental domain ${F}$, a generic point can be mapped to ${F}$ by an element of ${L_1}$

More generally, take two fundamental domains shared by two lattices. Example: the octapet, depending on one parameter ${s}$.

Theorem 2 Let ${R(s)=1-s}$ or ${frac(1/2s)}$ depending wether ${s<1/2}$ or ${s>1/2}$. If ${t=R(s)}$, then octapet${(t)}$ is a renormalization of octapet${(s)}$.

Consequences. The aperiodic set has positive codimension.

2.4. An algebraic family in ${{\mathbb C}^n}$

In ${{\mathbb C}^n}$, ${n=2,6,10,14,...}$, I define pairs of lattices associated to the ring ${{\mathbb Z}[i]}$. We call them complex octapets.

Theorem 3 Let ${e}$ be a composition of ${n}$ square turning maps for ${n=2,6,10,14,...}$. Then ${f=e^2}$ has an affine compactification from the algebraic family.

2.5. Outer billiards compactification

Theorem 4 Let ${P}$ be a polygon without parallel sides. Then the second return map to a strip admits, outside a bounded set, a double lattice compactification.

Experiments show that PETs are nearly renormalizable, but never exactly: tilings arose, but with mistakes.

3. Pseudo-group actions

Elements come with a domain. Make a graph whose vertices are points and edges join points mapped to each other by some element of the pseudo-group. The connected components are the orbits.

Now replace points by subsets. The maximal connected components are called tiles.

Example. The ${D_4}$ pseudogroup action. ${T}$ is the set of reflections in complexe coordinate hyperplanes in ${{\mathbb C}^2}$. Let ${h}$ be the involution which is a Hadamard matrix up to scale. Let ${S=T\cup hTh^{-1}}$. This pseudo-group contains the complex octapet.

Experiments show that every complex octapet tile is a union of pseudo-group tiles.

Corollary. The ${D_4}$ pseudo-group has an exact tiling by polytopes having ${D_4}$-symmetry. It is exactly renormalizable. This provides a description for the square turning map, which arises as a invariant plane in it.

The result on outer billiards requires some extra work.

## Notes of Richard Schwartz’ fifth Cambridge lecture 24-05-2017

The pentagram map and discrete integrable systems

Joint work with Valentin Ovsienko and Serge Tabachnikov

Start with a convex polygon. Draw diagonals between vertices at distance 2, they form a smaller polygon inside. Call this the pentagram map, although the number of sides is arbitrary.

The case of pentagons has been studied for centuries. Gauss’ Pentagrammon magnificum. Motzlan 1947.

In the non convex case, it becomes a projective construction (it is only generically defined). Hence it commutes with projective transformations. The space ${P_n}$ of ${n}$-gons up to projective transformation has dimension ${2n-8}$. The subset ${C_n}$ of convex polygons is homeomorphic to a ball (in fact, ${C_n}$ is a triangle bundle over ${C_{n-1}}$, as is seen by adding an extra vertex).

Let ${\phi:P_n\rightarrow P_n}$ denote the pentagram map.

Theorem 1 (Classical) If ${n=5,6}$, ${\phi}$ is periodic.

Let ${n=5}$. To a vertex, associate the cross-ratio of the sides and diagonals emanating from it. These numbers determine the polygon up to projective transformations. The cross-ratio of a vertex of ${\phi(P)}$ equals the cross-ratio of the 4 points along a diagonal, i.e. the cross-ratio of a vertex of ${P}$. Hence ${\phi^2=Id}$.

For ${n=6}$, I checked it using Mathematica.

For ${n\geq 7}$, no periodicity. Numerical experiments suggest that ${C_n}$ is foliated by ${\lfloor \frac{n-1}{2}\rfloor}$-tori, each with a natural flat structure. Whence the title. I will explain the origin of these tori.

1. Shrinking

Fact. The product of vertex cross-ratios ${\chi(P)}$ is ${\phi}$-invariant. Indeed, its log is the Hilbert length of ${\phi(P)}$ as a subset of ${P}$. View it as function of a point on a side. Complexify. Get a bounded holomorphic function on ${{\mathbb C}}$, it is constant. Checking on the regular polygon shows that ${\chi(\phi(P))=\chi(P)}$.

It implies that the diameter of ${\phi^j(P)}$ tends to 0.

Theorem 2 (Max Glick 2018) The shrink point is algebraic in the coordinates of the vertices.

Indeed, lift vertices to vectors in ${{\mathbb R}^3}$. Define an endomorphim of ${{\mathbb R}^3}$ by

$\displaystyle \begin{array}{rcl} T_P(A)=\sum_{i}\frac{det(V_{i-1},A,V_{i}}{det(V_{i-1},V_i,V_{i+1}}V_i, \end{array}$

Then ${T_{\phi(P)}=T_{P}}$.

A Poncelet polygon is a polygon which is inscribed in a conic and superscribed about a conic.

Poncelet polygons are fixed points of ${\phi}$.

2. Integrable systems

2.1. 4 dimensions

Let ${M}$ b a smooth closed 4-manifold, with a symplectic form ${\omega}$. To a smooth function ${f}$ on ${M}$, associate the Hamiltonian vector field ${H_f}$ such that ${\omega(H_f,V)}$. The flow of ${H_f}$ is tangent to the level sets of ${f}$.

Definition 3 Define the Poisson bracket by

$\displaystyle \begin{array}{rcl} \{f,g\}:=\omega(H_f,H_g). \end{array}$

Alternatively,

$\displaystyle \begin{array}{rcl} H_{\{f,g\}}=[H_f,H_g]. \end{array}$

Hence vanishing Poisson bracket means that the corresponding Hamiltonian flow commute. In non-degenerate cases, this implies that joint level curves are smooth surfaces with a natural map to ${{\mathbb R}^2}$. If the orbit of this action of ${{\mathbb R}^2}$ is closed, it must be a torus.

2.2. Higher dimensions

If, in dimension ${2n}$, ${n}$ functions Poisson-commute, one gets a map to ${{\mathbb R}^n}$ whose level sets are unions of orbits of an ${{\mathbb R}^n}$ action.

More generally, a Poisson bracket is a skew-symmetric map on vector fields, which is a derivation of the algebra of smooth functions and satisfies Jacobi identity.

I intend to turn ${P_n}$ into a symplectic manifold (in fact, merely indicate the Poisson bracket), in such a way that ${\phi}$ is symplectic. Then I will find ${n}$ Poisson-commuting functions.

F. Soloviev goes farther: he relates the dynamics of ${\phi}$ to Riemann surfaces, and so on.

3. Coordinates on ${P_n}$

Assign a cross-ratio to a flag, i.e. a vertex and an adjacent edge, ${\chi(e,v)=[u:v:w:x]}$ where ${u}$ is previous vertex, ${w}$ and ${x}$ are intersections of side ${e}$ with two next edges. This gives a map ${(I_1,\ldots,I_n):P_n\rightarrow{\mathbb R}^{2n}}$, this is too much for coordinates.

Define a twisted ${n}$-gon as a map ${\psi:{\mathbb Z}\rightarrow{\mathbb R} P^2}$ such that ${\psi(n+k)=M\psi(k)}$ for all ${k}$, for some projective transformation ${M}$. ${M}$ is called the monodromy. ${I_i}$‘s define coordinates on this enlarged space.

In terms of cross-ratio coordinates, ${trace(M)}$ and ${trace(M^{adj})}$ are rational functions, almost polynomials (only products of cross-ratios appear in the denominator).

3.1. Poisson structure

Define

$\displaystyle \begin{array}{rcl} \{I_0,I_2\}=I_0 I_2,\quad \{I_2,I_4\}=I_2 I_4,\quad \{I_1,I_3\}=-I_1 I_3,\quad \{I_3,I_5\}=I_3 I_5,\ldots \end{array}$

and all other Poissons brackets vanish. This defines a Poisson structure on twisted ${n}$-gons.

## Notes of Viktor Schroeder’s second informal Cambridge lecture 23-05-2017

Moebius structures on boundaries, II

1. Ptolemaic Moebius structures

Recall that a Moebius structure is Ptolemaic if cross-ratios satisfy the Ptolemaic inequality,

$\displaystyle \begin{array}{rcl} \rho_{12}\rho_{34}\leq \rho_{23}\rho_{14}+\rho_{13}\rho_{24}. \end{array}$

This means that ${crt}$ takes its values in the triangle ${\Delta\subset\hat\Sigma}$ with vertices at the extra points ${(1:1:0),(1:0:1),(0:1:1)}$.

Examples

1. Boundaries of ${CAT(-1)}$-spaces, in their Bourdon metric, are Ptolemaic.
2. Boundaries of hyperbolic groups admit natural Ptolemaic Moebius structures. This follows from a construction by Mineyev-Yu, as observed by Nica. Furthermore, Nica-Spakula observed that visual metrics associated to Green metrics associated to random walks also define Prolemaic Moebius structures.

There is an indirect link between Ptolemaic and triangle inequalities.

• Given a bounded Ptolemaic Moebius structure, pick a point ${\omega\in X}$ and form the semi-metric ${\rho_\omega}$ which sends ${\omega}$ at infinity. Then ${\rho_\omega}$ is a metric.
• Two metrics in the Moebius class which send the same point to infinity are proportional.
• A Ptolemaic Moebius structure always contains at least one bounded metric. Indeed, given any bounded semi-metric in the structure, add an ideal point ${\hat X=X\cup\{\hat\omega\}}$ et extend ${\rho}$ by ${\hat\rho(x,\hat\omega)=\rho(x,o)+1}$, where ${o}$ is an arbitrary origin in ${X}$. Then ${\hat\rho_{\hat\omega}}$ is a bounded metric in the Moebius structure of ${X}$.

2. The space of metrics of a Moebius structure

Given two Moebius-equivalent metrics ${\rho}$ and ${\tau}$, there exists a Lipschitz function ${\lambda}$ such that

$\displaystyle \begin{array}{rcl} \rho(x,y)=\lambda(x)\lambda(y)\tau(x,y). \end{array}$

One denotes by

$\displaystyle \begin{array}{rcl} \lambda=(\frac{d\rho}{d\tau})^{1/2}. \end{array}$

Question. Is every Moebius space the boundary of some space in a natural sense ?

This space should be the space ${\mathcal{M}}$ of metrics in the Moebius structure. The map

$\displaystyle \begin{array}{rcl} \rho\mapsto -\log\frac{d\rho}{d\tau}(x) \end{array}$

can be viewed as a Busemann function on ${\mathcal{M}}$. It is well defined up to an additive constant, since at each point,

$\displaystyle \begin{array}{rcl} \frac{d\rho}{d\tau}\frac{d\tau}{d\sigma}=\frac{d\rho}{d\sigma}. \end{array}$

Kingshook Biswas defines ${\mathcal{M}_a}$ as the subset of diameter 1, antipodal metrics in the Moebius structure. Antipodal means that every point has an other point lying at distance 1 from it. Then Biswas embeds ${\mathcal{M}_a}$ into ${C^0(X)}$ by ${\rho\mapsto -\log\frac{d\rho}{d\tau}}$. The image is a closed subset, ${\mathcal{M}_a}$ inherits a proper metric (balls are compact) that does not depend on the choice of reference metric ${\tau}$. He shows that, in the case of the boundary of a ${CAT(-1)}$ space ${Y}$, ${Y}$ isometrically embeds in ${\mathcal{M}_a}$, which is within bounded Hausdorff distance from ${Y}$.

3. Hausdorff measure

Assume that ${D}$ is the Haudorff dimension of ${(X,\rho)}$. Then Hausdorff measures corresponding to different metrics in the Moebius structures differ by a factor

$\displaystyle \begin{array}{rcl} \frac{d\mu_\rho}{d\mu_\tau}(x)=(\frac{d\rho}{d\tau}(x))^D. \end{array}$

Therefore the measure on pairs

$\displaystyle \begin{array}{rcl} d\nu(x,y)=\rho(y,s)^{-2D}d\mu_\rho(x)d\mu_\rho(y) \end{array}$

is invariant on ${X\times X}$.

Cross-ratio define a kind of distance on pairs.

4. Methods

4.1. Spheres between points

Given distinct points ${p,q,y\in X}$, the sets ${\{x\in X\,;\,[x:y:p:q]=1\}}$ form a 1-parameter family of “spheres” separating ${p}$ from ${q}$. Indeed, by the cocycle condition, this is an equivalence relation.

4.2. Jorgensen’s inequality

I learned this from J. Parker and S. Markham. Given a loxodromic transformation ${\alpha\in Moeb(X)}$, with axis ${p,q}$, set

$\displaystyle \begin{array}{rcl} a= \rho(p,q)\rho(z,\alpha z),\quad b=\rho(z,p)\rho(\alpha z,q),\quad c=\rho(\alpha z,p)\rho(z,q). \end{array}$

Then ${\frac{a}{b},\frac{b}{c},\frac{c}{a}}$ are cross-ratios. Define

$\displaystyle \begin{array}{rcl} m_\alpha=\sup_z \frac{a}{b}\frac{a}{c}. \end{array}$

Proposition 1 Let ${X}$ be a compact Moebius space. let ${\Gamma}$ be a discrete group of Moebius transformations of ${X}$. Then for every loxodromic elements ${\alpha,\beta\in\Gamma}$, ${d_\alpha}$ and ${d_\beta}$

$\displaystyle \begin{array}{rcl} m_\alpha^2(d((p,q),(\beta p,\beta q))+1)\geq 1. \end{array}$

5. Next time

I will explain Beyrer’s Moebius structure on the Furstenberg boundary of higher rank symmetric spaces.

## Notes of Nicolas Matte Bon’s Cambridge lecture 23-05-2017

Uniformly recurrent subgroups and rigidity of non-free minimal actions

Joint work with A. Le Boudec and T. Tsankov.

1. The Chabauty space of a group

Let ${G}$ be a locally compact group. The Chabauty space of ${G}$ is the set of subgroups of ${G}$, where two subgroups are nearby if their intersections with every compact set are. If ${G}$ is countable, the topology coincides with that induced from ${\{ 0,1 \}^G}$.

The ${G}$ action by conjugation on ${Sub(G)}$ is usually interesting. Glasner and Weiss suggested to study uniformly recurrent subgroups, aka URS, i.e. minimal invariant subsets of this action.

Examples.

1. Normal subgroups.
2. Conjigacy classes of cocompact subgroups.
3. Stabilizers of actions on compact spaces.

Indeed, the closure of the set of stabilizers of a ${G}$-action on ${X}$ contains a unique minimal subset, called the stabilizer URS of the action, ${S_G(X)}$ (Glasner-Weiss).

Theorem 1 (Matte Bon-Tsankov, Elek) Conversely, every URS of ${G}$ is the stabilizer URS of some action of ${G}$ on a compact space.

Our initial motivation was to study ${C^*}$ simplicity of countable groups, following this theorem.

Theorem 2 (Kalantar-Kennedy, Kennedy) If ${G}$is countable, the followng are equivalent.

1. ${G}$ is ${C^*}$-simple.
2. ${G}$ acts freely on its universal Furstenberg boundary.
3. ${G}$ has no non-trivial URS consisting of amenable subgroups.

Our interest has shifted to examples.

2. Examples: Thomson’s groups

These are three groups ${F acting respectvely on ${[0,1]}$, on the circle and on the Cantor set ${\{ 0,1 \}^{\mathbb N}}$. Each of them consists of all homeomorphisms acting locally like ${x\mapsto 2^n x+q}$. ${T}$ and ${V}$ act minimally (but ${F}$ does not), with large stabilizers.

Theorem 3 (Le Boudec-Matte Bon) Let ${T}$ act on a compact space ${X}$. Assume action is minimal and not topologically free. Then there is a continuous surjective equivariant map ${\phi:X\rightarrow S^1}$. Moreover, for all ${y\in S^1}$ but countably many, the action of the stabilizer of ${y}$ on the fiber ${\phi^{-1}(y)}$ is trivial.

Topologically free means that the set of points with trivial stabilizer is a dense ${G_\delta}$.

Free actions on the circle can be blown up: certain points are replaced with intervals. This gives mny examples.

Bounded cohomology method allow to show that certain group actions on the circle cannot be topologically free.

A similar statement holds fro ${V}$, and for ${F}$, it says that there are no such actions at all.

2.1. Proof

1. Classify all URS of ${T}$. There are only three: trivial, ${T}$ and the stabilizer URS of the action on the circle.
2. Construct a map to the circle in the third case.

The first step has a more general character. Let ${G}$ be a countable group acting faithfully on a compact Hausdorff space ${Z}$. We shall focus on subgroups fixing pointwise the complement of open sets ${U}$, ${G_U}$, called rigid stabilizers.

Proposition 4 Let ${H be a subgroup. Either a sequence of conjugates of ${H}$ converges to the trivial subgroup, or there exists an open subset ${U\subset Z}$ and a subgroup ${K stabilizing ${U}$ and whose action on ${U}$ coincides with that of a finite index subgroup of ${G_U}$. Moreover, if the the ${G}$ action on ${Z}$ is extremely proximal (every proper closed set can be shrinked to a point), then there exists a finite index subgroup of ${G_U}$ whose commutator subgroup is contained in ${H}$.

3. Non-discrete locally compact groups without URS nor IRS

This is work in progress.

3.1. Invariant random subgroups

An IRS is an invariant probability measure on Chabauty space. IRS are stabilizers of probability measure preserving actions.

Can a group have no IRS, or no URS, at all ? For instance, Neretin’s group is simple and has no lattices, this rules out the most obvious examples of IRS. Does it have others?

3.2. Answers in the discrete case

Tarski monsters have only countably many subgroups, so no URS nor IRS.

Finitary alternating group ${\mathfrak{A}_f({\mathbb N})}$ has no URS but plenty of IRS (Vershik).

Thompson’s groups ${T}$ and ${V}$ have 1 URS and no IRS (Dudko-Medynets).

The commutator subgroup of Thompson’s group ${F}$ has no URS, no IRS.

3.3. The non-discrete case

Compactly generated examples can be obtained, using dense subgroups in them obtained by slightly enlarging a known group. Neretin’s group is of this sort, starting from Thompson’s group ${V}$.

We describe non compactly generated examples. Let ${\Gamma}$ be a countable group action faithfully on a set ${\Omega}$. Let ${D_i}$ be finite subsets of ${\Gamma}$ with pairwise dsjoint supports, and such that for every ${g\in\Gamma}$, the support of ${g}$ has only finitely many points in common with any support${(D_i)}$. Consider the closed subgroup generated by ${\Gamma}$ and the product of ${D_i}$, in the topology where ${\prod D_i}$ is open.

Such constructions arise in Willis, Akim-Glasner-Weiss, Caprace-Cornulier to produce various counterexamples.

We take ${\Omega={\mathbb N}}$, ${\Gamma=\mathfrak{A}_f({\mathbb N})}$ and ${D_i=\mathfrak{A}([n_i,n_{i+1}])}$ for ${i}$ even, and trivial if ${n}$ s odd. Then the resulting group has no URS, but many IRS.

Let us take ${\Omega={\mathbb Q}_2}$, ${\Gamma=}$ the group of homeos locally modelled on ${x\mapsto 2^nx+q}$. It is a non-discrete variant of Thompson’s Then the resulting group has no IRS, but many URS.

## Notes of Erik Guentner’s Cambridge lecture 23-05-2017

Affine actions, cohomology and hyperbolicity

When can a group act properly on a Hilbert space or an ${L^p}$ space? I start from scratch.

1. Affine actions

${G}$ a discrete group, ${E}$ a Banach space. We are interested in actions of ${G}$ on ${E}$ where each element ${\alpha(g)}$ is an affine transformation whose linear part is isometric. Equivalently, one is given an linear representation ${\pi:G\rightarrow O(E)}$ and a cocycle ${b:G\rightarrow E}$, and

$\displaystyle \begin{array}{rcl} \alpha(g)=\pi(g)+b(g). \end{array}$

Such cocycles ${b\in Z^1(G,E)}$ represent classes in ${H^1(G,E)}$. Coboundaries give equivalent actions (merely change origin).

My point: typically, ${E=\ell_p(S)}$, ${S}$ a ${G}$-set, and coboundaries ${\phi-\pi_g(\phi)\in \ell_p(S)}$, for some ${\phi\in\ell_\infty(S)}$ which need not belong to ${\ell_p(S)}$.

The action is proper if ${|b(g)|\rightarrow\infty}$ as ${|g|\rightarrow\infty}$.

2. Hyperbolicity

2.1. Example: trees

${X=}$ simplicial tree. Let ${G}$ act on ${X}$. For vertex ${v}$, define ${\mu_v(x)\in\ell_1(lk(x))}$ as follows. If ${a}$ is a neighbour of ${x}$, set ${\mu_v(x)(a)=1}$ wether ${x}$ belongs to the arc ${[v,a]}$ or not.

This defines an action of ${G}$ on ${\ell_1(TX)}$, where ${TX}$ (the “tangent bundle of the tree”) is the union of all links of vertices, where the cocycle is

$\displaystyle \begin{array}{rcl} b(g)=\phi-g\circ\phi \end{array}$

where ${\phi=\mu_v}$, for a fixed ${v}$. Then ${\phi\in\ell_\infty(TX)}$. Nevertheless,

$\displaystyle \begin{array}{rcl} \|b\|_{\ell_1}=2 d(v,gv). \end{array}$

Hence it is proper.

2.2. Example: negatively curved manifolds

See the book by Nowak and Yu. ${\mu_v}$ become a unit vector field, the gradient of distance to ${v}$. The norm of ${\mu_v(x)-\mu_{gv}(x)}$ decays exponentially with ${d(x,v)}$ for fixed ${g}$ (in the tree case, it fell instantly from 1 to 0). If curvature is bounded below, balls grow exponentially, hence ${b(g)\in L^p(TM)}$. Every point in a tube around the geodesic segment between ${v}$ and ${gv}$ contributes to the norm, hence the ${L^p}$-norm is proportional to ${d(v,gv)}$, showing properness.

2.3. Example: hyperbolic groups

Inspired from Yu’s treatment. Let us define ${\mu_v(x)\in\ell_1(B(x,4\delta))\cap[x,v]_{2\delta}}$, where

$\displaystyle \begin{array}{rcl} [x,v]_{2\delta}=\{y\,;\,d(x,y)+d(y,v)\leq d(x,v)+2\delta\}. \end{array}$

What I describe corresponds to ${-\mu_v(x)}$ in the previous example: pointing towards ${v}$ instead of opposite to ${v}$. Move along a geodesic from ${x}$ to ${v}$. Iteratively average

Again, the key points are

• Bounded geometry.
• Exponential decay of information (Mineyev):

$\displaystyle \begin{array}{rcl} \|\mu_v(x)-\mu_w(x)\|_{\ell_1}\leq C\,e^{-\epsilon d(x,v)}. \end{array}$

3. Relative hyperbolicity

${G}$, ${P finitely generated groups. Define the combinatorial horoball over ${P}$ as follows. Start with infinitely many copies of ${Cay(P,S)}$ called layers, sitting in ${Cay(P\times{\mathbb Z},S\cup{1}). On }$n${-th layer, add edges between points at distance }$n$latex {. Attach such combinatorial horoballs over the cosets of }&fg=000000$P${ to }$Cay(G,S_G)${. The resulting graph }$X${ is }$\delta$latex {-hyperbolic. Unfortunately, is does not have bounded geometry, this kills Mineyev estimates. \nbegin{theorem}{Guentner-Reckwerdt-Tessera} Assume }&fg=000000$P${ has polynomial growth. Then }$G${ admits a proper affine action on a mixed }$\ell_p-\ell_1${-space, and also on an }$L^p$latex { space. \end{theorem} In Yu’s treatment, passing from }&fg=000000$\ell_p-\ell_1${ to }$L^p$latex { follows from bounded geometry. \subsection{Proof} The new trick has independent interest. \begin{enumerate} \item Exponential growth of balls with center in the base. \item Existence of thinned horoballs. \item Averaging technique. \end{enumerate} By a thinned horoball, I mean a bounded geometry subset }&fg=000000$\theta${ of the combinatorial horoball, obtained by decimating vertices in slices. It is still quasi-isometric to the combinatorial horoball. Unfortunately, the }$P${-action is lost. It is thinned with respect to a choice of parameters. Let }$\Theta${ denote the set of thinned horoballs with fixed parameters. It is compact in pointed Gromov-Hausdorff topology. Hence there is a }$P${-invariant probability measure. A thinning of the whose cusped space is a choice of thinned horoball in each glued combinatorial horoball }$(\theta_t)_{t\in G/P}${, }$\theta_t\in \Theta${. The infinite product of }$P${-invariant measure is a }$G\$-invariant probability measure in the space of thinnings. Apply the method of Mineyev-Yu, Lafforgue to each thinning and then average.

4. Questions

Schwartz: you could weight the edges instead? We tried but could not make it work.

Breuillard: what is the growth

## Notes of David Kyed’s Cambridge lecture 18-05-2017

${\ell^2}$-Betti numbers of universal quantum groups

Joint with Pichon, Arndt, Vaes,…

I spoke on the same subject in the same room in 2006. I think my understading has improved.

1. Infinite discrete groups

Let ${\Gamma}$ act on vectorspace ${X}$. There are cohomology groups ${H^p(\Gamma,X)}$. 1-cocycles are maps ${c:\Gamma\rightarrow X}$ such that

$\displaystyle \begin{array}{rcl} c(\gamma\mu)=\gamma c(\mu)+c(\gamma). \end{array}$

1-coboundaries are ${\gamma\mapsto \gamma\xi-\xi}$, ${\xi\in X}$.

This coincides with Hochschild cohomology of the group algebra ${{\mathbb C}\Gamma}$ for the module ${X_\epsilon}$.

When ${X=\ell^2\Gamma}$, what is ${H^p(\Gamma,\ell^2\Gamma)}$ ? This is either 0 or infinite dimensional. One can attach a finite dimension to it: view it as a ${L\Gamma}$-module and take its ${dim_{L\Gamma}}$. What is this?

Assume that ${\Gamma}$ is finitely generated. Then cocycles are determined by their values on a finite generating set

$\displaystyle \begin{array}{rcl} (c(\gamma_1),\ldots,c(\gamma_n))\in \ell^2\Gamma^n. \end{array}$

It is a closed subspace, so one can project orthogonally to it. The projector is an ${n\times n}$ matrix with coefficients in ${L\Gamma}$, hence it has a finte trace. This defines

$\displaystyle \begin{array}{rcl} \beta_1^{(2)}(\Gamma)=\mathrm{dim}_{L\Gamma}\Gamma,\ell^2\Gamma)-1. \end{array}$

What are ${\ell^2}$-Betti numbers good for?

If ${\Gamma}$ is amenable, all ${\ell^2}$-Betti numbers vanish (Cheeger-Gromov). For free groups, it depends on rank.

The algebraic conjectures, like Kadison-Kaplansky’s, often follow from deeper ${\ell^2}$-Betti numbers conjectures.

2. Discrete quantum groups

A discrete quantum group is a Hopf, unital, star-algebra. It has a Haar trace ${h}$, which need not be a usual trace. We shall focus on quantum groups of Kac type, for which ${h}$ is a trace. This rules out quantum ${SU(2)}$.

Unfortunately, quantum groups tend not to act on spaces. The GNS construction allow to define an analogue of ${L^2({\mathbb C}\Gamma,h)}$, denoted by ${\ell^2\mathbb{G}}$, hence homology groups. When ${h}$ is a trace, ${\ell^2}$-Betti numbers are well defined.

2.1. Examples

Start with a compact Lie group (viewed as a dual of an inexistant discrete group). For ${O_n}$, let ${{\mathbb C}\hat O_n^+}$ be the ${\star}$ algebra generated by ${n}$ elements ${v_{ij}}$ subject to ${vv^\perp=1=v^\perp v}$.

For ${U_n}$, let ${{\mathbb C}\hat U_n^+}$ be the ${\star}$ algebra generated by ${n}$ elements ${u_{ij}}$ subject to ${u}$ and ${\bar u}$ being unitary.

For the symmetric group, viewed as permutation matrices, a similar algebra ${\hat S_n^+}$.

In all cases, the algebra can be view as a freed version of the group algebra. ${{\mathbb C}\hat U_n^+}$ is the non-commutative analogue of free groups. It behves much the same (it is a ${II_1}$ factor, it has rapid decay,…).

Open question. Is ${L\hat U_n^+\sim \hat U_m^+}$ for ${n\not m}$ ?

Banica proved that ${L\hat U_2^+\sim LF_2}$. ${\ell^2}$-Betti numbers have proved efficient in distinguishing these algebras.

3. Results

Vergnioux 2008 proved that ${\beta_1^{(2)}\hat O_n^+=0}$.

Collins-Hartel-Thom proved that ${\beta_p^{(2)}\hat O_n^+=0}$. This implies that ${\hat O_n^+}$ is not isomorphic to ${LF_m}$.

Theorem 1 (Raum 2016, Bichon-Kyed-Raum 2017) ${\beta_p^{(2)}\hat U_n^+=0}$ except for ${p=1}$ where it is equal to 1.

Vaes-Popa-Shlyakhtenko have defined ${\ell^2}$-Betti numbers for tensor categories. Such have representation categories ${Rep(\hat{\mathbb{G}})}$.

Theorem 2 (Kyed-Raum-Vaes-Valrekens 2017) ${\beta_p^{(2)}(\hat{ \mathbb{G}})=\beta_p^{(2)}(Rep (\hat{\mathbb{G}}))}$.

This might be the right definition, since the right hand side is always defined, whereas the left hand side makes sense only when ${h}$ is a trace.

4. Questions

What values can ${\ell^2}$-Betti numbers take ?

Tim Austin proved that uncountably many real numbers were ${\ell^2}$-Betti numbers. Since, Lukasz Grabowski (student of Schick) proved that every constructible real number is an ${\ell^2}$-Betti number.

## Notes of Nina Friedrich’s Cambridge lecture 17-05-2017

Homological stability of moduli spaces of high-dimensional manifolds

1. Manifolds

Say a sequence manifolds and maps satisfies homological stability if induced maps on homomology groups become eventually isomorphisms.

How does one prove such a property? We are interested in the special case of diffeomorphim groups. Assume these manifolds are groups. Find a highly connected complex ${X}$ with action of ${X_n}$, then use a Serre spectral sequence.

Theorem 1 Let ${2n\geq 6}$, let ${W}$ be a ${2n}$-manifold with non-empty boundary, and with virtually polycyclic fundamental group. Consider the map

$\displaystyle \begin{array}{rcl} BDiff_\partial(W)\rightarrow BDiff_\partial(W\#(S^n\times S^n)) \end{array}$

induced by surging in ${(\partial W\times I)\#(S^n\times S^n)}$. Then this map is an isomorphism in homology ${H_k}$ for ${k}$ bounded above by ${\frac{1}{2}(g(W)-h(\pi_1(W))-5)}$.

Here, ${h}$ is the Hirsch length and ${g}$ the genus defined as follows:

Definition 2 The genus of a manifold is the maximum num of copies of ${S^n\times S^n}$ whose connected sum (with a disk deleted) can be embedded in ${W}$.

Let ${I^{tr}_n(W)}$ be the set of immersions of ${S^n}$ in ${W}$ with trivial normal bundle, together with a path from base point of W to image of base point of the sphere. This is the ${n}$-th homotopy group of ${n}$-frames in ${W}$. It has the structure of an abelian group. Furthermore, ${I^{tr}_n(W)}$ is a ${{\mathbb Z}[\pi_1(M)]}$-module.

The intersection form defines a ${{\mathbb Z}[\pi_1(M)]}$-valued bilinear form ${\lambda}$ on ${I^{tr}_n(W)}$. It counts a loop, obtained from paths from base-point, for each intersection point of spheres. It is skew-hermitian (note that the group ring has an involution).

Finally, there is a map ${\mu:I^{tr}_n(W)\rightarrow{\mathbb Z}[\pi_1(M)]/\Lambda}$ that counts Modding out by ${\Lambda=im(\lambda)}$ is necessary to make ${\mu}$ linear, since otherwise ${\mu(a+b)=\mu(a)+\mu(b)+\lambda(a,b)}$.

Equipped with ${\lambda,\mu}$, ${I^{tr}_n(W)}$ is a “quadratic module” in the sense of Wall.

A hyperbolic module is ${{\mathbb Z}[\pi_1(M)]^2}$ with basis ${(e,f)}$ satisfying ${\mu(e)=\mu(f)=0}$ and ${\lambda}$ has matrix ${\begin{pmatrix} 0 & 1 \\ (-1)^n & 0 \end{pmatrix}}$. This is the quadratic module of a punctured product of two spheres.

Define ${HU(M)}$ as the space of semi-simple quadratic modules mod those containing ${k+1}$ copies of a hyperbolic module.

Our topological theorem follows from the following algebraic result.

Theorem 3 Under the same assumptions as Theorem 1, ${HU(I^{tr}_n(W))}$ is ${\frac{1}{2}(g(W)-h(\pi_1(W))-6)}$-connected.

In fact our theorem is more general, involving an invariant of rings ${R}$, the unitary stable rank ${usr(R)}$, instead of Hirsch length, and an invariant of quadratic modules, ${g(M)}$, the Witt index of ${M}$, is the maximal number of copies of the hyperbolic module whose direct sum can be embedded in ${M}$.

Say a sequence ${v_i}$ in ${M}$ is unimodular, if there exist ${f_i:R\rightarrow M}$ and ${\phi_i:M\rightarrow R}$ such that ${f_i(1)=v_i}$ and ${\phi_j\circ f_i=\delta_i^j \,1_R}$. Say a ring ${R}$ satisfies ${(S_n)}$ if for every ${n}$-term unimodular sequence ${r_i}$, there exists scalars ${t_i\in R}$ such that ${(r_1+t_1r_{n+1},\ldots,r_n+t_nr_{n+1})}$ is again unimodular. Say ${R}$ has unitary stable rank ${n}$, ${usr(R)=n}$, if ${n}$ is minimal such that ${(S_n)}$ and ${(T_{n+1})}$ hold. I skip the definition of ${(T_n)}$.

Example. Fields have ${usr=1}$ and integers have ${usr=2}$ (the Euclide algorithm proves ${(S_2)}$).

3. Proof

${\lambda}$-unimodularity is the variant where linear forms ${\phi_i}$ are required to be of the form ${\lambda(w_i,\cdot)}$. For sums of hyperbolic modules, this makes no difference, but it does in general. This is the main new difficulty compared to previously existing results.

4. Question

Is the theorem true for all groups? Yes. We are able to estimate ${usr}$ only in the polycyclic case.