Notes of Marc Burger’s third Leverhulme lecture 24-04-2017

Towards higher Teichmuller theory, III

1. On integer points in the Hitchin moduli space

Let {G} be a simple Lie group of real split type ({Sp(n,{\mathbb R}),Sl(n,{\mathbb R}),SO(n,n+1)}…). Then {Hom(\Gamma_g,G)} has a Hitchin component. In the sequel, we focus on {G=PSl(n,{\mathbb R})}. The Hitchin component is the connected component which contains the Fuchsian representations

\displaystyle  \begin{array}{rcl}  \mathcal{F}_n(S_g)&=&\{\pi_n\circ\rho_h\,;\,\rho_h:\Gamma_g\rightarrow PSl(2,{\mathbb R})\textrm{ orientation preserving holonomy} \\ &&\textrm{ representation of a hyperbolic structure }h\textrm{ on }S_g\}. \end{array}

where {\pi_n:PSl(2,{\mathbb R})\rightarrow PSl(n,{\mathbb R})} is the irreducible representation.

We denote by {Hit_n(S_g)} its image in

\displaystyle  \begin{array}{rcl}  Rep(\Gamma_g,PSl(n,{\mathbb R}))=Hom(\Gamma_g,PSl(n,{\mathbb R}))^{ss}/PSl(n,{\mathbb R}). \end{array}

Examples.

  1. If {n=2}, {Hit_2(S_g)} has only Fuchsian representations, it is Teichmüller space, of dimension {6g-6}.
  2. If {n=3}, {Hit_3(S_g)} is the space of marked convex projective sutructures on {S_g} (convex means that the universal convering space is a convex open set in {{\mathbb R} P^2}.

Facts.

  1. {Hit_n(S_g)} has dimension {(2g-2)\mathrm{dim}(Sl(n,{\mathbb R}))} (Hitchin 1992).
  2. Mapping class group {Mod_g} acts properly discontinuously on it (Labourie 2006).
  3. Every Hitchin representation has an open domain of discontinuity in

1.1. Classical ideas

Many features of Teichmüller theory generalize:

The Weil-Petersson metrics generalizes into the Pressure metric (Bridgemann-Canary-Labourie-Sambarino).

The Bonahon-Thurston shear coordinates can be generlized too (Bonahon-Dreyer).

Thurston-Penner coordinates have been generalized by Fock-Gontcharov.

The Collar Lemma has been generalized by Zhang-Lee.

1.2. New features

However, new phenomena show up:

Entropy rigidity behaves differently (Potrie-Sambarino).

Integer points become of interest.

2. Integral representations

Definition 1 Let {\rho:\Gamma_g\rightarrow Sl(n,{\mathbb R})} be a representation.

  1. Say {\rho} is integral if all traces are integers.
  2. Given a lattice {\Lambda<Sl(n,{\mathbb R})} be a lattice. Say {\rho} is {\Lambda}-integral if {\rho(\Gamma_g)<\Lambda}.

Let {Hit_n^{\mathbb Z}(S_g)}, {Hit_n^\Gamma(S_g)} denote the sets of such representations.

These are {Mod_g}-invariant subsets.

Question. Is {Mod_g\setminus Hit_n^\Gamma(S_g)} finite ? Is {Mod_g\setminus Hit_n^{\mathbb Z}(S_g)} finite ?

2.1. Dimension 2

This turns out to be the case for {n=2}. Nevertheless, the two cases are inequivalent. The proof of the second statement is much harder.

Example. Let {a,b} be square-free integers. Let {H^{a,b}} denote the corresponding quaternion algebra (i.e. the 4-dimensional {{\mathbb Q}}-algebra with basis {1,i,j,k} and relations {i^2=a,j^2=b,ij=-ji=k}). Then unit elements form a group isomorphic to {Sl(2,{\mathbb R})}, integer units form a surface group. Its genus is given by a complicated formula. All these examples, and all their finite index subgroups, give points of

\displaystyle  \begin{array}{rcl}  \bigcup_{g>1}Hit_2^\mathbb Z(S_g). \end{array}

However, modulo {Mod_g}, there are only finitely many points with given genus. This is by no means obvious from the genus formula.

2.2. Dimension 3

Theorem 2 (Long-Reid-Thislethwaite) If {n=3}, {Mod_g\setminus Hit_3^{\mathbb Z}(S_g)} is infinite.

This is a by-product of their rational parametrization of {Hit_3}. Applied to a 1-parameter family of representations of a {(3,3,4)} triangle group, it produces Zariski-dense subgroups which are pairwise non commensurable. The open convex sets on which they act are projectively inequivalent, they converge to a triangle. This has to do with the fact that there is a limiting action on a triangle building.

2.3. Higher dimensions

The sequel is ongoing work with Francois Labourie and Anna Wienhard.

Proposition 3 For all {n\geq 4}, {Mod_g\setminus Hit_n^{\mathbb Z}(S_g)} is infinite.

This does not follow from the 3-dimensional case. Indeed, a 3D representation, viewed as a 4D representation, is not Hitchin.

Since this set is infinite, we want to measure its growth in terms of some notion of height.

Recall that the translation length of {g\in Sl(n,{\mathbb R})} when acting on the symmetric space is equal to the Euclidean norm of the vector of {\log} of absolute values of eigenvalues of matrix {g}. Given two representations {\rho} and {\pi}, set

\displaystyle  \begin{array}{rcl}  A(\rho,\pi)=\sup_{\gamma\in\Gamma_g}\frac{\ell(\rho(\gamma))}{\ell(\pi(\gamma))}. \end{array}

For {n=2}, this is called Thurston’s asymmetric distance.

Fact. If {\pi} is Fuchsian, {\ell(\pi(\gamma))} is bounded below by the conjugacy length of {\gamma} (min of length if conjugates with respect to a word metric).

Here is our notion of height: it measures distance to Fuchsian representations.

Definition 4

\displaystyle  \begin{array}{rcl}  h(\rho)=\inf_{\pi\in\mathcal{F}_n(S_g)}A(\pi,\rho). \end{array}

Then {h} is a {Mod_g}-invariant function.

Theorem 5 (Burger-Labourie-Wienhard) The number of Hitchin representations of height {\leq T}, mod the mapping class group, is finite.

The minimal area of a representation is the infimal area of an equivariant map to the symmetric space. We can show that the number of Hitchin representations of minimal area {\leq T}, mod the mapping class group, is finite. This is clear for {\Lambda}-integral representations, since then equivariant maps descend to maps to a fixed compact of finite volume manifold.

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Notes of Brian Bowditch’s Cambridge lecture 20-04-2017

Bounding genera of singular surfaces

Ultimate motivation: understand the curve complex for non-compact surfaces. But today, only closed surfaces {\Sigma} around.

1. Genus distance

The curve complex {\mathcal{C}} of {\Sigma} is hyperbolic, pseudo-Anosov diffeos act loxodromically, moving vertices at a linear speed.

Say {\alpha\sim\beta} if there is a compact surface {S} with {\partial S=\alpha_0\cup \beta_0} and a map {S\rightarrow\Sigma} mapping {\alpha_0} to {\alpha} and {\beta_0} to {\beta}. Let {A=A(\alpha)} denote the equivalence class of {\alpha}. There are two cases. Either {\alpha\sim0}, i.e. {\alpha} is a separating curve. Or {\alpha\not\sim 0}.

Fact. {A} is 3-dense in {\mathcal{C}} (1-dense in the separating case).

Give, {\alpha,\beta\in A}, define {\rho(\alpha,\beta)=} minimal genus of surface {S} achieving {\alpha\sim\beta}. This is a metric. Morally, this is related to commutator length.

Question. Is this genus distance comparable to the distance in {\mathcal{C}}?

2. Pseudo-Anosov distorsion

Example. Let {\phi} be a pseudo-Anosov diffeo such that {h(\alpha)\sim\alpha} (such maps exist). Recall that {d(\alpha,\phi^n\alpha)\sim n}. I claim that {\rho(\alpha,\phi^n\alpha)\sim n}.

The proof requires 3-manifold topology. Let {g=\rho(\alpha,\phi^n\alpha)}, achieved by some surface {S} with a map {f:S\rightarrow\Sigma}. If {\gamma\subset S} is an essential simple closed curve, then {f(\gamma} is not null homotopic in {\Sigma} (otherwise, one could surge {S} into a surface of lower genus). Let {M_\phi} be the mapping torus. It is hyperbolic. Let {M\rightarrow M_\phi} be the cyclic cover, diffeomorphic to {\Sigma\times{\mathbb R}}, with periodic geometry, let {\psi} be the deck transformation. Let {\alpha^*} be the closed geodesic in {M} freely homotopic to {\alpha}. In {M},

\displaystyle  \begin{array}{rcl}  d(\alpha^*,\psi^n\alpha^*)\sim n. \end{array}

Thurston-Bonahon realise the composed map {F:S\rightarrow\Sigma\rightarrow M} by a 1-Lipschitz map from {S} equipped with a hyperbolic structure with concave boundary (it amounts to triangulating {S}). Note that Area{(S)\leq 2\pi(2g+1)} is linear in {g}. Also, the injectivity radius of {S} is bounded from below independently on {n}. So the diameter of {S} is linear in {g}. Thus

\displaystyle  \begin{array}{rcl}  d(\alpha^*,\psi^n\alpha^*)\leq C\,g, \end{array}

and {g\geq c\, n}. However, {c} depends on {\phi} and {\alpha}. Note that diameter{(\alpha^*)} is bounded.

3. Result

Theorem 1 There is a constant {L}, depending only on {\Sigma}, such that for all {\alpha,\beta\in A},

\displaystyle  \begin{array}{rcl}  d(\alpha,\beta)\leq L\,\rho(\alpha,\beta). \end{array}

The proof is similar. Let {f:S\rightarrow\Sigma} minimize the genus {g} of {S}.

Fact. There exists a complete hyperbolic 3-manifold {M}, homeomorphic to {\Sigma\times{\mathbb R}}, with arbitrarily short representatives {\alpha^*} and {\beta^*}.

It can be taken quasi-Fuchsian. Again, there exists a 1-Lipschitz map {F:S\rightarrow\Sigma\rightarrow M}, where {S} is hyperbolic with concave boundary, hence linear area. However, the injectivity radius of {M} is not controlled. The thin part of {S} is mapped to the thin part of {M}, a union of solid tori. Let us electrify tubes: change to a metric which is zero on the thin part. Then {F} remains 1-Lipschitz. In the electrified metric, the diameter of {S} is bounded by its area, so the distance between {\alpha^*} and {\beta^*} in the electrified metric is linear in {g}.

Fact. {d(\alpha,\beta)\leq L\,d_{elec}(\alpha^*,\beta^*)}.

This is a consequence of the Ending Lamination Theorem. One uses the quasi-isometric model of {M}. The constant {L} depends only on {\Sigma}, but it is not effective.

Question (H. Wilton). Stable commutator length is more natural than commutator length. Would replacing {\alpha} and {\beta} by powers simplify the argument ? I do not see how it could help.

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Notes of Andreas Aaserud’s Cambridge lecture 20-04-2017

Property (T) and approximate conjugacy of actions

Joint work with Sorin Popa.

We let countable groups act on probability spaces. We focus on free and ergodic actions.

Example. Bernoulli action on {X^\Gamma}.

1. Actions on von Neumann algebras

Von Neumann algebras have states (linear functionals which are nonnegative with valu

Example. {L^\infty(X,\mu)} has a state, defined by measure {\mu}.

Actions on standard probability spaces are in 1-1 correspondence with state preservong actions on separable abelian von Neumann algebras.

Ergodicity (no invariants but multiples of 1) and freeness (the largest projection fixed by all automorphisms fixes group elements) generalize to actions on von Neumann algebras.

There is a distance on automorphisms: sup of difference on elements of {L^\infty}-norm less than one, in {L^2}-norm.

Definition 1 Say two state-preserving actions are approximately conjugate if there exists a sequence of state-preserving {\star}-isomorphisms {\theta_n} such that the {\theta_n}-conjugate of one action tends to the second action pointwise.

Ornstein-Weiss-Popa: all free ergodic measure-preserving actions of an amenable group are mutually approx. conjugate.

This is reminiscent of orbit equivalence, although there is no logical relation between aprox. conjugacy and orbit equivalence.

Proposition 2 Every group with an infinite amenable quotient has infinitely many non conjugate actions which are approx. conjugate.

We use Bernoulli actions, and the fact they are classified by entropy.

Theorem 3 For Kazhdan groups, appr. conjugacy implies conjugacy.

2. Property (T)

Theorem 3 follows from

Lemma 4 Let {\Gamma} have property (T), let {S} be a finite generating set. There exists {\delta>0} such that if two actions are {\delta}-close, then they are conjugate.

The proof of the Lemma goes as follows.

  1. Construct a von Neumann algebra {M}, generated by {A} and one operator for each group element, which encodes the action.
  2. To the inclusion {A\subset M}, Vaughan Jones associates the basic construction {\mathcal{M}}, generated by {M} and by a projection {e_A} of {M} onto {A}. In admits a trace, hence a Hilbert space {H} of elements of {\mathcal{H}} with finite Hilbert-Schmidt norm.
  3. From the two given actions of {\Gamma} on {A}, cook up an action of {\Gamma} on {H} with {e_A} almost fixed (it uses one action on the left and the other on the right). A fixed vector close to {e_A} provides a projection {e} in {\mathcal{M}}.
  4. Show that {e} is of the form {e=ve_A v^*} where {v} normalizes {A}. Then {Ad(v^*)} is the desired automorphism.

2.1. Standard representation

The trace gives rise to a Hilbert space {L^2(M,\tau)}, wth a left-regular representation.

2.2. Cartan inclusions

It is {A\subset M} where {A} is maximal abelian in {M} and {M} is generated by the normalizer of {A}.

Let {L^2(A)} be the closure of {A} in {L^2(M)}. It comes with an orthogonal projection {e_A:L^2(M)\rightarrow L^2(A)}. The right-regular version of {L^2(A)} and {L^2(A)} generate a subalgebra {\mathcal{A}}.

Elements of the form {xe_A y} generate a dense subalgebra of {\mathcal{M}}.

2.3. Feldman-Moore

This a construction in ergodic theory. The full group of an action is made of maps which coincide piecewise with group elements. This notion generalizes to actions on von Neumann algebras.

The von Neumann algebra {M} is the cross-product constructed from the full group of the action. It comes with a trace, and {A\subset M} is a Cartan inclusion. Going to the full group is necessary in order that {A} be maximal abelian in {M}.

3. Open problems

Do non-Kazhdan groups have approx. conjugate non-conjugate actions ?

We can prove that non-amenable groups admit at least 2 non aprox. conjugate actions. Do all of them have infinitely many ?

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Notes of Emmanuel Breuillard’s fifth Cambridge lecture 19-04-2017

Introduction to approximate groups, V

Today, I explain mathematics from the 1950’s: the solution of Hilbert’s 5th problem. Indeed, ideas from this old theory enters our structure theory of approximate groups.

1. Hilbert’s 5th problem

In 1900, Hilbert wondered wether one can remove differentiability from the theory of Lie groups. Before him, Lie had shown that {C^2} Lie groups are the same as analytic Lie groups. At that time, groups were always thought of as acting on some manifold, so Hilbert asked wether a continuous action of a manifold must be differentiable.

1.1. 1951

A Bourbaki report by Serre in 1951 (a year before the problem was solved) identifies two problems:

  1. A locally Euclidean topological group is a Lie group.
  2. A locally compact subgroup of homeomorphisms of a manifold is a Lie group.

(2) implies (1). (1) is now a theorem by Montgomery-Zippin, based on work by Gleason.

Problem (2) is still open (sometimes called Hilbert-Smith conjecture). It is known in dimension 2 (Montgomery-Zippin) and 3 (Pardon). It is known for Lipschitz homeomorphisms.

Problem (2) is equivalent to wether {{\mathbb Z}_p} can act faithfully by homeomorphisms on a manifold.

1.2. 1952

In 3 Annals papers, Gleason proves that an NSS group is a Lie group. NSS = no small subgroups.

Montgomery-Zippin showed that locally Euclidean groups are NSS. Also, they solved problem (2) for transitive actions: if a locally compact group acts faithfully and transitively on a locally connected finite dimensional topological space, then it is NSS.

1.3. 1953

Yamabe proved a structure theorem for locally compact groups.

Theorem 1 (Yamabe) Every locally compact group {G} has an open subgroup {G'} which is approximated by Lie groups: every neighborhood {U} of the identity in {G'} there exists a normal subgroup {K} of {G'}, contained in {U}, such that {G'/K} is Lie with finitely many components.

In 1955, Montgomery-Zippin wrote a book account of these results. Nowadays, Yamabe’s theorem rather directly implies all others.

1.4. Previous results

Haar (1930) constructed the Haar measure. Cartan-von Neumann showed that closed subgroups of Lie groups are Lie.

Peter-Weyl proved the compact case of Yamabe’s theorem. They show that

\displaystyle  \begin{array}{rcl}  L^2(G)=\bigoplus_{\pi\in\hat G}\mathcal{H}_\pi \end{array}

where {\mathcal{H}_\pi} is finite dimensional. It follows that {G} embeds in a (countable product of) general linear groups. It is based on the spectral theorem. Indeed, construct a continuous function {\phi} with small support. Make it inversion and conjugation invariant. Show that the operator of convolution with {\phi} is compact (Arzela-Ascoli). Hence it has a sequence of finite dimensional eigenspaces. The spectral theorem says that

\displaystyle  \begin{array}{rcl}  L^2(G)=\bigoplus_{\lambda}\mathcal{H}_\lambda \end{array}

Projection to finitely many summands gives a homorphism to {Gl(N)} with small kernel.

The abelian case follows from Pontrijagin’s duality theory.

Gelfand-Raikov tried to use representation theory in the noncompact case as well, but their efforts did not have a posterity.

2. Sketch of Yamabe {\Rightarrow} Montgomery-Zippin

Note that there exist connected compact groups which are not locally connected: the solenoid {{\mathbb R}\times{\mathbb Q}_2/{\mathbb Z}[\frac{1}{2}]}.

Let {G} act faithfully transitively on a locally connected and finite dimensional topological space {X}. Up to taking an open subgroup, by Yamabe, one can assume that {G=\lim G_n} where {X_n=G_n/H_n} and {G_n} is Lie. {X_n} is a manifold, and {X} is a projective limit of {X_n}. Since {X} is finite dimensional, dimension stabilizes, {X_n} is a finite covering space of {X_{n_0}}, hence {H_n} is profinite. By local connectedness of {X}, {H_n} is finite, {G} is Lie.

2.1. Isometry groups

Note that Gromov uses a slightly different version, dealing with isometry groups. It follows easily from Yamabe’s theorem.

3. Proof of Yamabe’s Theorem

Yamabe’s theorem is a local statement. Hence versions for local groups appeared. Jacoby’s version from the 1950’s was flawed. Isa Goldbring finally produced a correct proof in 2010. This is the result of decades of works which enlightened the fact that the proof goes smoothly in the language of nonstandard analysis (Hirschfeld in the 1970’s) and model theory (van den Dries and Goldbring’s seminar notes, appeared in L’Enseignement Mathématique). Tao’s 2013 book gives a modern treatment.

3.1. Gleason-Yamabe lemmas

Let {G} be a locally compact group. Let {U} be a neighborhood of 1. For {g\in U}, let

\displaystyle n_U(g)=\sup\{n\in{\mathbb N}\,;\,1,g,\ldots,g^n\in U\}.

Analogously for subsets {Q\subset U},

\displaystyle n_U(Q)=\sup\{n\in{\mathbb N}\,;\,1,Q,\ldots,Q^n\subset U\}.

The escape norm is the inverse

\displaystyle  \begin{array}{rcl}  \|g\|_U=\frac{1}{n_U(g)}. \end{array}

Lemma 2 Assume that {G} is NSS. Let {U,V} be sufficiently small compact neighborhoods of 1. Then

  1. For all compact sets {Q\subset G}, {\frac{1}{C}n_U(Q)\leq n_V(Q)\leq C\,n_U(Q)}.
  2. For all {g,h\in U},

    \displaystyle  \begin{array}{rcl}  \|hgh^{-1}\|_U&\leq& C\|g\|_U.\\ \|gh\|_U&\leq& C(\|g\|_U+\|h\|_U).\\ \|[g,h]\|_U&\leq& C\|g\|_U\|h\|_U. \end{array}

This is easy for Lie groups, using differentiability properties of the exponential maps. If (1) fails, then extracting subsequences, one would obtain a nontrivial subgroup contained in {U}. In his 1951 survey, Serre alludes to the fact that if (2) would hold, one would be finished. More precisely, he means the following lemma, used in the proof of Lemma 1.

Lemma 3 Let {G} be a locally compact group. Let {V} be a neighborhood of 1. Then there exists {C} such that for all {K\geq 1}, there exists a neighborhood {U\subset V} of {1} such that for all compact subsets {Q\subset U},

  • either {n_{V^4}(Q)\geq K\,n_V(Q)},
  • or {\exists g\in Q} such that {n_U(g)\leq KC\,n_V(Q)}.

This lemma, combined with the Peter-Weyl Theorem, implies approximability of locally compact groups by NSS groups. Indeed, it has the following

Corollary 4 (Subgroup trapping) For all {V} there exists {U\subset V} such that the subgroup generated by all subgroups of {G} contained in {U} is contained in {V}.

3.2. Proof of Gleason’s theorem

Gleason’s theorem NSS {\Rightarrow} Lie uses Lemma 1 as follows. If {G} is NSS, then every element has a unique square root. Going to limits shows that every element belongs to some 1-parameter subgroups. The set of 1-parameter subgroups is a candidate for a Lie algebra. Addition is defined by

\displaystyle  \begin{array}{rcl}  X(t)+Y(t)=\lim_{n\rightarrow\infty}(X(\frac{t}{n})Y(\frac{t}{n}))^{n}. \end{array}

Lemma 1 tells that the {n}-th power stays bounded. The bracket estimate implies that the limit exists. A locally compact vectorspace is finite dimensional. The adjoint action is well defined, it maps {G} to a Lie group, with a kernel which is abelian, hence Lie.

3.3. Proof of Lemma 1

As is Peter-Weyl’s Theorem, pick a bump function {\phi}. Let

\displaystyle  \begin{array}{rcl}  \partial_g \phi(x)=\phi(g^{-1}x)-\phi(x). \end{array}

Then trivially

\displaystyle  \begin{array}{rcl}  \|\partial_{gh} \phi\|_\infty\leq \|\partial_g \phi\|_\infty+\|\partial_h \phi\|_\infty. \end{array}

Also {\|\partial_g \phi\|_\infty<1\Rightarrow g\in U}. This implies that {\|g\|_U\leq 2\|\partial_g \phi\|_\infty}. The key point is to get the reverse inequality {\|\partial_g \phi\|_\infty\leq C\|g\|_U}.

If {\phi} were {C^2}, Taylor’s expansion

\displaystyle  \begin{array}{rcl}  \partial_{g^n} \phi=n\partial_{g} \phi+\sum_{i=0}^{n-1}\partial_{g^i}\partial_{g} \phi \end{array}

would prove it. In absence of regularity, Gleason uses convolution. For instance, if {\phi=\psi_1\star\psi_2}, a second derivative

\displaystyle  \begin{array}{rcl}  \partial_g\partial_h\phi=\int_{G}\partial_g\psi_1(y)\partial_{yhy^{-1}}\psi_2(y^{-1}x)\,dy \end{array}

is expressed in terms of first derivatives.

These techniques still work for locally compact local groups, and allow to conclude that any locally compact local group has the same germ as a group. This fails in infinite dimensions.

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Notes of Mikolai Fraczyk’s Cambridge lecture 19-04-2017

Benjamin-Schramm convergence of arithmetic 3-manifolds

1. B-S convergence and limit multiplicity

Let {G} be a semi-simple Lie group, let {\Gamma_i} be a sequence of lattices in {G}, without infinite repetitions. We say that locally symmetric spaces {\Gamma_i\setminus X} B-S converge if the relative volume of the thin part tends to 0. The limit is then a probability distribution on the set (topologized by Gromov-Hausdorff convergence) of finite volume locally symmetric spaces.

In their 7-author paper, Abert-Bergeron-… showed that

  1. If {G} has higher rank ans has property (T), then for every non-trivial sequence, the limit is {X}.
  2. For general semisimple {G}, given an arithmetic lattice, the relative volume of the thin parts of its congruence coverings is at most a negative power of the volume. Thus B-S convergence holds.

1.1. Limit multiplicity

If {\Gamma<G} is a lattice, {G} acts on {L^2(\Gamma\setminus G)}. If {\Gamma} is uniform, this unitary representation {R_\Gamma} splits as a direct sum of irreducibles {\pi\in \hat G} with multiplicities {m_\Gamma(\pi)}. We define the atomic measure on {\hat G}

\displaystyle  \begin{array}{rcl}  \mu_\Gamma:=\frac{1}{\mathrm{Vol}(\Gamma\setminus G)}\sum_{\pi\in \hat G}m_\Gamma(\pi)\delta_\pi, \end{array}

Definition 1 Let {(\Gamma_i)} be a sequence of uniform lattices in {G}. Say that it has the limiting multiplicity property if {\mu_{\Gamma_i}} converges to the Plancherel measure on {\hat G}.

Sauvageot (1996) gave the following characterization. L-M property holds iff for every smooth compactly supported function {f} on {G},

\displaystyle  \begin{array}{rcl}  \lim_{i\rightarrow\infty}\frac{\mathrm{Trace}(R_{\Gamma_i}(f)}{\mathrm{Vol}(\Gamma_i\setminus G)}=f(1). \end{array}

Abert-Bergeron et al showed that if injectivity radii are uniformly bounded from below, B-S convergence implies limiting multiplicity property. The reason is that injectivity radius at the base-point tends to infinity, thus the effect of convolution with a small support kernel becomes independant of the kernel.

The converse is true (folklore): limiting multiplicity property implies B-S convergence.

2. Result

I complete the discussion of congruence lattices in the remaining 2 and 3-dimensional cases.

2.1. Torsion free uniform case

Theorem 2 Let {G=PGl(2,{\mathbb R})} or {PGl(2,{\mathbb C})}. Let {\Gamma} be a congruence uniform torsion free arithmetic lattice. Then

  1. Thin parts have small volume,

    \displaystyle  \begin{array}{rcl}  \mathrm{Vol}((\Gamma\setminus G)_{<R})<_R \mathrm{Vol}(\Gamma\setminus G)^{1-\delta}, \end{array}

    where constants depend only on {R}. Note that “congruence” is intrinsically defined.

  2. Limiting multiplicity property holds. Given smooth compactly supported function {f} on {G} such that {\|f\|_\infty\leq 1}, support in {B(1,R)}, then

    \displaystyle  \begin{array}{rcl}  |\frac{\mathrm{Trace}(R_{\Gamma}(f))}{\mathrm{Vol}(\Gamma\setminus G)}-f(1)|<_R\mathrm{Vol}(\Gamma\setminus G)^{-\delta}. \end{array}

  3. In the non-congruence arithmetic case, the estimate on the thin part involves the discriminant of the trace field (the field generated by traces of matrices {Ad_\gamma}, {\gamma\in\Gamma},

    \displaystyle  \begin{array}{rcl}  \mathrm{Vol}((\Gamma\setminus G)_{<R})<_R \mathrm{Vol}(\Gamma\setminus G)\Delta_{k(\Gamma)}^{-\delta}. \end{array}

In fact, (2), applied to a smoothed characteristic function of a ball implies (1).

Odlyzko’s bound indicates that discriminants are usually large:

\displaystyle  \begin{array}{rcl}  \Delta(k)>60\#\{\textrm{real places}\}\times 22\#\{\textrm{complex places}\}. \end{array}

Corollary 3 For {\Gamma} arithmetic in {Sl(2,{\mathbb C})}, {M=\Gamma\setminus H^3} admits a triangulation {T} with {O(\mathrm{Vol}(M))} vertices, degrees of vertices {\leq 245}.

This had been conjectured by Gelander. There are many small triangles in the thin part, but their number is overwhelmed by large triangles of the thick part. The corollary relies on Dobrowolski’s 1976 theorem on Mahler measures of algebaric numbers, which yields a lower bound on injectivity radius

\displaystyle  \begin{array}{rcl}  injrad(M)>(\log[k(\Gamma):{\mathbb Q}])^{-3}. \end{array}

{T} is obtained as the nerve of a cover of {M} by balls.

2.2. Torsion case, non-uniform case

(Joint work with Jean Raimbault). We prove that every sequence of arithmetic congruence lattices of {PGl(2,{\mathbb R})} or {PGl(2,{\mathbb C})} B-S converges to {X}.

2.3. Proof of torsion case, non-uniform case

Follow the 7 author paper. Consider the invariant random subgroup {\nu_\Gamma} associated to {\Gamma} (uniform measure on conjugates in the Chabauty space of closed subgroups of {G}. The 7 show that B-S convergence is equivalent to convergence of the IRS to the Dirac measure at the trivial subgroup. We show that any limit of {\nu_{\Gamma_i}} is supported on subgroups containing only torsion by unipotent elements. Such a subgroup cannot be Zariski dense. The 7 show that every nontrivial IRS is Zariski dense.

3. Proof of main theorem

The goal is the estimate

\displaystyle  \begin{array}{rcl}  |\frac{\mathrm{Trace}(R_{\Gamma}(f))}{\mathrm{Vol}(\Gamma\setminus G)}-f(1)|<_R\mathrm{Vol}(\Gamma\setminus G)^{-\delta}. \end{array}

Using Selberg’s trace formula,

\displaystyle  \begin{array}{rcl}  \mathrm{Trace}(R_{\Gamma}(f))=\sum_{\mathrm{conjugacy\,classes}\,[\gamma]}\mathrm{Vol}(\Gamma_\gamma\setminus G_\gamma)\int_{G_\gamma\setminus G}f(x^{-1}\gamma x)\,dx. \end{array}

The dominant term comes from the trivial conjugacy class, one must estimate all other terms. If support{(f)\subset B(1,R)}, the number of nonzero terms is the number of closed geodesics of length {<R}, i.e. elements with eigenvalues {< e^R}. These eigenvalues {\lambda} are algebraic integers of a special type. Indeed, uniform arithmetic lattices are integer matrices in a product of one copy of {Sl(2)} and a number of orthogonal groups. Therefore, among the Galois conjugates of {\lambda}, at most two can have modulus {\not=1}. The Weil height {h(\lambda)} is {O(1/[{\mathbb Q}[\lambda]:{\mathbb Q}])}, it is very small. Equidistribution results exist for numbers of small Weil height.

Theorem 4 (Bilu) If {z_i} are algebraic numbers of degrees tending to infinity and Weil height tending to 0, then the Galois conjugates of {z_i} get uniformly distributed on the unit circle.

This implies a bound on nonzero terms,

\displaystyle  \begin{array}{rcl}  \mathrm{Vol}(\Gamma_\gamma\setminus G_\gamma)<\Delta_{k(\Gamma)}^{1/2}e^{o([k(\Gamma):{\mathbb Q}])}, \end{array}

\displaystyle  \begin{array}{rcl}  |\int_{G_\gamma\setminus G}f(x^{-1}\gamma x)\,dx|<e^{o([k(\Gamma):{\mathbb Q}])}. \end{array}

and also on the number nonzero terms, when combined with a result of Kabatianskii-Levenstein on the number of nearly orthogonal unit vectors in Euclidean space.

Theorem 5 (Kabatianskii-Levenstein, Tao) Let {v_1,\ldots,v_m\in {\mathbb R}^n} be unit vectors that satisfy

\displaystyle  \begin{array}{rcl}  |v_i\cdot v_j|\leq \frac{A}{n}. \end{array}

Then {m\ll n^{CA}}, for some absolute constant {C}.

Indeed, one constructs a unit vector in {{\mathbb R}^{[k(\Gamma):{\mathbb Q}]-2}} for each characteristic polynomial of element of length {<e^R}. Bilu’s theorem implies that they are nearly orthogonal.

Borel’s volume formula gives

\displaystyle  \begin{array}{rcl}  \mathrm{Vol}(\Gamma\setminus G) \sim \Delta_{k(\Gamma)}^{3/2}\frac{\zeta_{k(\Gamma)}(2)}{(4\pi^2)^{[k(\Gamma):{\mathbb Q}]}}> \Delta_{k(\Gamma)}^{1/2+\epsilon} \end{array}

thanks to Odlyzko’s bound.

This gives the result for maximal lattices. Generalizing to suitable finite index subgroups requires the notion of congruence subgroup.

Let {A} be a quaternion algebra over a number field {k}. Let {G} be the group of unit quaternions. Then {G(k\otimes{\mathbb R})} is a product of {Sl(2,{\mathbb R})}‘s, {Sl(2,{\mathbb C})}‘s and {SO(3)}‘s. Adjusting parameters allows to have exactly one noncompact factor. Pick an order {\mathcal{O}} of {A}, i.e. an {\mathcal{O}_k}-submodule which is a subring. Then {\mathcal{O}\cap G} is a lattice in {G(k\otimes{\mathbb R})}, which projects to a lattice of {Sl(2)}. The previous arguments apply to such lattices, provided {\mathcal{O}} is maximal. Say a lattice {\Gamma} in {Sl(2)} is congruence if it contains the projection of {\{x\in\mathcal{O}\cap G\,;\, x\equiv 1\mod N\mathcal{O}\}} for some order {\mathcal{O}} and some integer {N}.

To treat the case of a congruence lattice {\Gamma<\Gamma_{max}}, one views {L^2(\Gamma\setminus G)} as sections of a vectorbundle over {\Gamma_{max}\setminus G} associated with representations of {Sl(2,{\mathbb Z}/N{\mathbb Z})}.

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Notes of Karen Vogtmann’s Cambridge lecture 13-04-2017

RAAG subgroups of RAAGs

Joint work with Beatrice Pozzetti.

RAAGs have plenty of interesting subgroups. Weird Bestvina-Brady beasts. Every 3-manifold group virtually embeds in a RAAG.

Carl Droms proved that if graph {\Gamma} does not contain a length 3 path or a 4-cycle as a full subgraph, then every finitely generated subgroup of the RAAG {A_\Gamma} is a RAAG.

What about other RAAGs?

1. Motivation

I am interested in (outer) automorphism groups of RAAGs. Among interesting examples are free and free abelian groups. They are dratsically different. For instance, {Out(F_n)} is not linear (Formanek-Procesi), whereas {Out({\mathbb Z}^n)} is. {Out(F_n)} has property FA (fixed points on trees, in fact, {{\mathbb R}}-trees) (Culler-Vogtmann). However, a finite index of {Out(F_3)} acts on a tree (it maps onto {Sl(2,{\mathbb Z})}, Lubotzky), so it is not Kazhdan.

Question. Does {Out(F_n)}, {n\geq 3}, have property (T) ?

Here is Lubotzky’s argument. Consider the morphism of {F_3} to {{\mathbb Z}/2} kiling two generators, let {K} be its kernel. Consider the subgroup {Aut(F_3,K)} sending {K} to {K}. This has finite index in {Aut(F_3)}. {{\mathbb Z}/2} acts on the double cover of the bouquet of 3 circles that correspinds to {K}. This graph has connectivity 5. So {Aut(F_3,K)} acts on {{\mathbb Z}^5}, and commutes with the {{\mathbb Z}/2}. The {{\mathbb Z}/2}-action has a {{\mathbb Z}^2} eigenspace, whence an action of {Aut(F_3,K)} on {{\mathbb Z}^2}.

Grunewald-Lubotzky generalized this. Given a finite quotient {Q} of {F_n}, with finite index kernel {K}, the centralizer {Aut^Q(F_n,K)} in the action on the homology of covering graph preserves eigenspaces.

Question. Analogously, can you find interesting representations of {Aut(A_\Gamma)}?

The answer depends on the structure of graph {\Gamma}. There is a partial ordering on (equivalence classes) vertices: say {v\leq w} if {lk(v)\subset st(w)}. Say {v\sim w} if {v\leq w} and {w\leq v}. Equivalence classes split among abelian and non-abelian classes (depending wether they commute or not).

Proposition 1 (Charney-Vogtmann) There is a finite index subgroup {Out^0(A_\Gamma)<Out(A_\Gamma)} and epimorphisms

\displaystyle  \begin{array}{rcl}  E_v:Out^0(A_\Gamma)\rightarrow Out^0(A_{\Gamma\setminus[v]}),\quad R_v:Out^0(A_\Gamma)\rightarrow Out^0(A_{[v]}) \end{array}

if {[v]} is maximal. If {[v]} is not maximal, need to compose {E_w} with one last {R_v}.

So, if {[v]} is abelian or if {|[v]|\leq 3}, we get actions on trees.

Grunewald-Lubotzky’s argument uses induction based on the fact that subgroups of free groups are free. This raises the question wether finite index normal subgroups of RAAGs are RAAGs.

2. Diameter of a finite quotient

Build a graph {\Gamma_0} whose vertices are equivalence classes of vertices of {\Gamma}. It embeds in {\Gamma}.

Let {\pi:A_\Gamma\rightarrow Q} finite be an epimorphism, {K=\mathrm{ker}(\pi)}. Let {d_\pi} be the maximal distance in {\Gamma_0} between equivalence classes which are mapped to nontrivial elements of {Q}.

{d_\pi=0} means that everything is mapped to 1 but elements of one single equivalence class. In this case, {K} is a RAAG. Indeed, use the Salvetti complex {S_\Gamma} (union of tori). {\pi} descends to {A_{[v]}}. The corresponding covering merely opens the bouquet of circles corresponding to {A_{[v]}} into a graph.

Let us assume that {d_\pi=1} and {\Gamma} is a tree. Then {\Gamma_0} is a tree. Let {v} and {w} be two vertices representing distinct classes, not mapped to 1.

Proposition 2 If {\langle \pi v\rangle \cap \langle \pi w\rangle=\{1\}}, then {K} is a RAAG. Otherwise

  • either {v} or {w} is a leaf. Then {K} is a RAAG.
  • Otherwise, {K} is not a RAAG.

On can see that {A_\Gamma} is not always coherent.

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Notes of Indira Chatterji’s Cambridge lecture 13-04-2017

Spaces admitting thin triangles and median spaces

Joint work with Cornelia Drutu and Frederic Haglund, initiated 10 years ago.

Theorem 1 Let {X} be a space with walls. Assume that {X} is {\mu}-locally finite. Assume that the wall metric has thin triangles. Then the median space associated to the wall structure is at finite distance Hausdorff distance from {X}.

Corollary 2 Lattices in groups {SO(n,1)} and in products of copies of such act isometrically and properly, with bounded quotient, on a median space.

10 years ago, we hoped to deduce an action on a {CAT(0)} cube complex. It turns out this is not the case: with Fernos and Iozzi, we proved that cocompact lattices in {SO(n,1)\times SO(n,1)} are not cubulable (by bounded cohomology superrigidity). So the corollary is the best one can expect.

1. Media spaces and spaces with walls

In a geodesic metric space, the interval between two points is

\displaystyle  \begin{array}{rcl}  I(x,y)=\{z\in X\,;\,d(x,z)+d(z,y)=d(x,y)\}. \end{array}

Say that {X} is median if for every triangle, the three intervals intersect at a single point {m}. {m} is called the median.

Examples. {{\mathbb R}}, {{\mathbb R}}-trees, {\ell^1}, {L^1}, {CAT(0)}-cube complexes with {\ell^1}-metric on cubes.

Definition 3 A measured wall space is a set {X} equipped with a collection {\mathcal{H}} of subsets (called half-spaces) stable under complementation, with a measure {\mu} on {\mathcal{H}}, such that the measure of the subset of half-spaces separating two points is always finite.

Examples. Hyperplanes in {CAT(0)}-cube complexes with counting measure. All hyperplanes in {{\mathbb R}^n} with the natural motion-invariant measure. All hyperplanes in hyperbolic space {H^n} with the natural motion-invariant measure.

Definition 4 Given a measured wall space, defined the wall (pseudo-)metric

\displaystyle  \begin{array}{rcl}  d(x,y)=\mu(w(x|y)). \end{array}

Example. In {{\mathbb R}^n} or {H^n}, one recovers the Euclidean (resp. hyperbolic) metric.

Proposition 5 Any wall metric embeds isometrically in a median space.

Indeed, consider sections, i.e. maps {s} assigning to each wall one of the 2 a half-spaces it bounds. Say {s} is admissible if disjoint walls are never mapped to opposite half-spaces.

Example. Fix {x_0\in X}. The section {s_{x_0}} which associates to a wall the half-space containing {x_0} is admissible.

Let {\overline{M(X)}} be the set of admissible sections. Let {M(X)} be the subset of sections {s} such that {s\Delta s_{x_0}} has finite measure. Then {M(X)} inherit a metric, it is median.

2. Spaces admitting thin triangles

Say a metric space admits thin triangles if {\exists \delta} such that for any 3 points {x,y,z}, there exist {x',y',z'} on intervals such that diameter{(x',y',z')\leq\delta}.

Example. Hyperbolic metric spaces and groups, median spaces, {\ell^1} products of those. However, Euclidean spaces do not admit thin triangles.

Example. A tripod of {\ell^1} planes has thin triangles but it is not median.

Not much is known about such spaces. They are expected to have quadratic filling, but only a sub-cubic bound is known.

3. Sketch of proof

Assume {X} is a space with walls. Assume that the measure of the set of walls intersecting a ball is bounded above, uniformly in terms of its radius. Assume that {X} admits thin triangles. We prove that {M(X)} is at finite Hausdorff distance of the image of {X}.

Here is the key step.

Lemma 6 Let {C\subset X} be a convex set. Let {x\in X}, let {p} be the projection of {x} to {C} ({\epsilon}-projection suffices). Then

\displaystyle  \begin{array}{rcl}  \mu(w(x|p)\setminus w(x|C))\leq 2\delta+\epsilon. \end{array}

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