## Notes from the workshop Boundaries of groups and representations, Wien, 22-24 fevrier 2016

Workshop Boundaries of groups and representations, Vienna, february 22-24, 2016

1. Alexander Lubotzky: Arithmetic quotients of the mapping class group

Joint work with Fritz Grunewald (GAFA 2008, automorphism group of free groups), Michael Larsen, Justin Malestein (GAFA 2015).

Let ${A=Out(F_g)}$, ${M_g=}$ mapping class group, ${M}$ means either one of them. ${A}$ maps onto ${Gl(g,{\mathbb Z})}$ and ${M_g}$ maps to ${Sp(2g,{\mathbb Z})}$, these are arithmetic quotients, but there are many more.

Theorem 1 For every pair ${(H,\rho)}$ where ${H}$ is a finite group with ${d(H) generators and ${\rho}$ an irreducible ${{\mathbb Q}}$-representation of ${H}$, there exists an explicitely given arithmetic group ${\Gamma(H,\rho)}$ and a virtual epimorphism ${\phi:M\rightarrow\Gamma}$, where virtual means that ${\phi}$ is defined on a finite index subgroup of ${M}$ and is onto a finite index subgroup of ${\Gamma}$.

The classical case correspond to ${H=\{e\}}$.

1.1. Construction

If ${H}$ is a finite group and ${\pi:F_g\rightarrow H}$ is an epimorphism, ${R=\mathrm{ker}(\pi)}$, if ${R/R'}$ is viewed as an ${H}$-module, then the ${{\mathbb Q}[H]}$-module ${{\mathbb Q}\otimes(R/R')}$ is isomorphic to ${{\mathbb Q}[H]^{g-1}\oplus{\mathbb Q}}$ (Gashitz). A similar result holds for surface groups ${T_g=\pi_1(S_g)}$: ${{\mathbb Q}\otimes(R/R')}$ is isomorphic to ${{\mathbb Q}[H]^{2g-1}\oplus{\mathbb Q}^2}$ (Chevalley-Weil).

Write ${{\mathbb Q}[H]}$ as a sum of simple ${{\mathbb Q}}$-algebras

$\displaystyle \begin{array}{rcl} {\mathbb Q}[H]={\mathbb Q}\oplus\bigoplus_{i=1}^\ell A_i, \end{array}$

with ${A_i=M_{n_i}({D_i}_{|k_i})}$, where ${k_i}$ is a finite extension of ${{\mathbb Q}}$ and ${D_i}$ a central division algebra.

In both cases, define

$\displaystyle \begin{array}{rcl} M(\pi)=\{\alpha\in M\,;\,\pi\circ\alpha=\pi\}. \end{array}$

This is the set of automorphisms preserving ${\pi}$ (viewed as a finite index subgroup in ${Aut}$ rather than in ${Out}$ for the moment). Then ${M(\pi)}$ preserves ${R}$ and ${R/R'}$ as an ${H}$-module. Since ${{\mathbb Q}\otimes(R/R')}$ has a ${{\mathbb Q}^{g-1}\oplus {\mathbb Q}={\mathbb Q}^g}$ (resp. ${{\mathbb Q}^{2g}}$) summand, we get a morphism ${M(\pi)\rightarrow Gl(g,{\mathbb Q})}$ (resp. ${Sp(2g,{\mathbb Q})}$). I am interested in the other summands ${A_i^{g-1}}$ (resp. ${A_i^{2g-2}}$).

1.2. Main technical theorem

The image of ${M(\pi)}$ in ${Aut_{A_i}(A_i^{g-1})}$ preserves a lattice, since we tensored a ${{\mathbb Z}}$-module into a ${{\mathbb Q}}$-module. Hence it is contained in the ${\mathcal{O}_i}$ points of a ${k_i}$-algebraic group ${\mathbb{G}_i}$. In the mapping class group case, the modules ${A_i^{g-1}}$ inherit a bilinear form arising from cup-products in ${T_g}$.

1.3. Examples

If ${\pi:F_g}$ or ${T_g\rightarrow H}$ is degenerate, meaning, in the free group case, mapping one of the basis to the identity

Let us start with the simplest example ${H=C_2={\mathbb Z}/2{\mathbb Z}}$. Then ${R}$ has index 2 in ${F_g}$, ${R/R'}$ has rank ${2g-1=g+(g-1)}$, whence 2 representations, one, the classical one, to ${Gl(g,{\mathbb Z})}$, and a new one, to ${Gl(g-1,{\mathbb Z})}$. For a while, I thought the image was nilpotent, this would have solved the irritating question wether ${Aut(F_g)}$ has property (T). But I was wrong, we get a virtual epimorphism of ${Out(F_g)}$ to ${Sl(g-1,{\mathbb Z})}$.

For ${g=3}$, we get a virtual epimorphism to ${F_2}$. I think this should not indicate that ${Out(F_g)}$ does not have property (T) for higher ${g}$.

Let ${H=C_p}$. Then ${{\mathbb Q}[C_p]={\mathbb Q}\oplus k_p}$ where ${k_p}$ is the cyclotomic field. This yields an epimorphism of ${Aut(F_g)}$ to ${Gl(g-1,{\mathbb Z}[\sqrt[p]{1}])}$.

There is some theory about which division algebras arise as summands in group algebras (the Schur subgroup of the Brauer group). This helps. Here, we get a virtual epimorphism of ${Out(F_g)}$ to ${SL_{m(g-1)}({\mathbb Z})}$ for each ${m}$ using ${H=\mathfrak{S}_m \times C_p}$.

The mapping class group case requires dealing with sesquilinear forms over noncommutative rings. The involution arises from ${h\mapsto h^{-1}}$ on ${{\mathbb Q}[H]}$. Over the reals, there are only three central divison algebras, ${{\mathbb R}}$, ${{\mathbb C}}$ and ${\mathbb{H}}$, the arithmetic groups that we encounter are ${C_n}$ types ${Sp(2m(g-1),{\mathbb Z})}$, ${Sp(4m(g-1),\mathcal{O})}$, ${\mathcal{O}}$ the ring of integers of ${{\mathbb Q}(\sqrt[p]{1})}$, ${A_n}$ types ${SU(m(g-1),m(g-1),\mathcal{O}')}$ where ${\mathcal{O}'={\mathbb Z}[\sqrt[p]{1}]}$, and ${D_n}$ types ${SO(2m(g-1),2m(g-1),\mathcal{O}')}$… but we found no type ${B_n}$.

1.4. Proofs

The free group case could be handled more or less by hand, using Fox calculus.

The mapping class group case is harder. The handlebody subgroup (Hamenstädt) of the mapping class group provides an action of ${Aut(F_g)}$ in the mapping class group picture, and this helped a lot.

1.5. Questions

What are the kernels of these epimorphisms ? What is their intersection ?

2. Ludovic Marquis: projectivization of some Dehn fillings on hyperbolic 4-orbifolds

Joint work with Suhyoung Choi and Gye-Seon Lee.

2.1. Coxeter groups

We deal with Coxeter groups. Given a graph with labels ${m_{ij}\in\{3,4,..,\infty\}}$, consider the presentation ${\sigma_i^2}$, ${(\sigma_i\sigma_j)^2}$ if no edge joining ${i}$ to ${j}$, ${(\sigma_i\sigma_j)^{m_{ij}}}$ otherwise. This produces a group ${W}$.

Coxeter, Vinberg: if ${W}$ is irreducible (connected graph), then ${W}$ is either finite, either virtually ${{\mathbb Z}^d}$ (both these cases are classified) or pretty large.

2.2. Andreev’s theorem

Let ${P}$ be 3-dimensional polyhedron. Label each edge ${e}$ with an angle ${\theta_e\in)0,\frac{\pi}{2}]}$. Andreev’s theorem realizes ${P}$ as a convex hyperbolic polyhedron with dihedral angles ${\theta_e}$, in a unique manner (up to hyperbolic motions). If ${\theta_e}$ are divisors of ${2\pi}$, reflections in the faces of ${P}$ generate a Coxeter group. Faces of ${P}$ correspond to vertices of the Coxeter graph, certain edges of ${P}$ to edges of the graph.

2.3. Dehn filling

What if one deforms ${P}$ by collapsing an edge ? The Coxeter graph persists. Let label the vanishing edge with angle ${\pi/m}$. All Coxeter groups ${W_m}$ are quotients of ${W_\infty}$, hence representations ${\rho_m:W_\infty\rightarrow Isom(\mathbb{H}^3)}$. As ${m}$ tends to infinity, ${\rho_m}$ converge to ${\rho_\infty}$.

This behaviour never arises in higher dimensions.

Theorem 2 (Garland-Raghunathan) For all ${d\geq 4}$, for all lattices ${\Gamma, local rigidity holds: every representation of ${\Gamma}$ in a neighborhood of the obvious one is conjugate to it.

Nevertheless, we shall perform Dehn filling in projective geometry.

2.4. Reflection groups in projective geometry

Here is a projective variant of Poincaré’s theorem on hyperbolic reflection groups.

Theorem 3 (Tits-Vinberg) Let ${P}$ be a polytope of ${{\mathbb R} P^d}$. For each facet ${s}$ of ${P}$, pick a reflection ${\sigma_s}$ fixing ${s}$ pointwise, in such a way that each time faces ${s}$ and ${t}$ intersect, ${\sigma_s \sigma_t}$ is the direct sum of identity on ${s\cap t}$ and a 2-dimensional rotation in a complementary plane. Then

1. ${\sigma_s}$ generate a Coxeter group ${\Gamma}$ with a discrete and faithful action ${\rho}$ on projective space.
2. ${\Gamma\cdot P}$ is a convex subset of projective space.
3. The interior ${\Omega}$ of ${\Gamma\cdot P}$ is tiled by the ${\gamma(P\cap\Omega)}$.
4. Theaction of ${\Gamma}$ is proper with ${P\cap\Omega}$ as a fundamental domain.
5. Assume that ${W}$ is irreducible and large, and that the action of ${\Gamma}$ on ${{\mathbb R}^{d+1}}$ is irreducible. Then ${\Omega}$ is properly convex.

2.5. Projective Dehn filling

We study an example of a graph ${\mathcal{G}}$, an edge of which is labelled ${m>6}$. ${\mathcal{G}}$ describes a polyhedron which is the product of two triangles.

Theorem 4 For ${m}$ finite, there are two polytopes realizing ${\mathcal{G}}$. For ${m=\infty}$, there is only one, congruent to a hyperbolic polytope.

It is standard that, for such examples,

1. For finite ${m}$, ${\Gamma_m}$ acts cocompactly on ${\Omega_m}$. ${\Omega_m}$ is not tructly convex, its boundary is not ${C^1}$.
2. For ${m=\infty}$, ${\Omega_\infty}$ is an ellipsoid and ${\Omega_\infty/\Gamma_\infty}$ has finite volume.

In our examples, ${\Omega_m}$ contains properly embedded triangles ${\Delta}$ (i.e. ${\partial \Delta\subset\partial\Omega_m}$) but no higher dimensional tetrahedra (unlike previously known similar examples by Benoist).

2.6. Questions

Extreme points in the boundary ?

What about attaching handles in other intermediate dimensions ?

Bounds on dimensions of projective reflection groups ? In principle, such a bound should exist.

3. Gye-Seon Lee: Collar Lemma for Hitchin representations

The Collar Lemma for hyperbolic syrfaces (Linda Keen) implies that if two curves intersect, there is a lower bound for the length of one in terms of the length of the other, independently of the hyperbolic structure on the surface. We generalize this to other families of representations.

3.1. Hitchin representations

Let ${C(S)}$ be the space of conjugacy classes of representations ${\pi_1(S)\rightarrow PSL(3,{\mathbb R})}$ which are discrete, faithful and cocompact on a convex open subset of projective plane. I.e. the space of convex real projective structures on ${S}$.

Let ${Hit_n(S)}$ be the Hitchin component. Then ${Hit_2=}$ Teichmüller space and ${Hit_3=C(S)}$.

Labourie : Hitchin representations are discrete and faithful, and every element is mapped to a diagonalizable matrix whose eigenvalues have distinct absolute values. We denote by ${\ell_\rho(\gamma)=\log(\frac{\lambda_n}{\lambda_1})}$. Then ${\ell_\rho}$ equals hyperbolic length of closed geodesics if ${n=2}$ and the Hilbert length of closed geodesics if ${n=3}$.

3.2. Results

Linda Keen’s result states that a short geodesic has a wide tubular neighborhood which is a topological annulus. It implies that if ${\eta}$ and ${\gamma}$ have nonzero intersection number ${i(\eta,\gamma)}$, then

$\displaystyle \begin{array}{rcl} \sinh(\frac{\ell(\eta)}{2i(\eta,\gamma)})\sinh(\frac{\ell(\gamma)}{2})\geq 1. \end{array}$

Theorem 5 (Zhang-Lee) Let ${\rho}$ belong to the Hitchin component. Same statement holds, where ${\sinh}$ is replaced with ${\exp -1}$.

This does not hold for all Anosov representations.

Example 1 Quasi-Fuchsian representations in ${PSl(2,{\mathbb C})}$ form a product of 2 copies of Teichmüller space. By Epstein-Marsden-Markovic, ${\ell_\rho}$ is bounded above by twice the min of the lengths of coordinates, which contradicts a Collar Lemma.

Linda Keen’s result is sharp (consider punctured tori). Recently, Nicolas Tholozan obtained a sharp generalization for ${n=3}$. Our result is not sharp.

Corollary 6 For a Hitchin representation, there are at most ${3g-3}$ primitive closed curves with length ${<\log 2}$.

3.3. Proof

If ${n=3}$, this is nice projective geometry. Ratios of lengths are interpreted as cross ratios of 4-tuples of points or lines. The key step is to control the order of points (axes of group elements and their images) along the ideal boundary.

In general, we use Labourie’s realization of the ideal boundary of ${\pi_1(S)}$ as a completely positive curve in the flag manifold. Notions like being on the positive side of a flag, or cross-ratios geenralize to the flag manifold.

Tholozan proves that every convex real projective structure is dominated by a Fuchsian one, in the sense that the length function is uniformly smaller. We do not know wether a sharp bound can hold for higher ${n}$.

4. Joan Porti: Geometry and dynamics of Anosov representations I

Joint work with Misha Kapovich and Bernhard Leeb.

4.1. Anosov representations

The notion appears for surface groups and ${PSl(n,{\mathbb R})}$, in his study of Hitchin’s component (2006).

The general definition, for hyperbolic groups and semi-simple Lie groups, is due to Olivier Guichard and Anna Wienhard (2012).

Definition 7 Let ${X=G/K}$ be a symmetric space of noncompact type, ${B}$ a Borel subgroup, ${F=G/B}$. Let ${\Gamma}$ be a hyperbolic group, let ${\rho\in Hom(\Gamma,G)}$. A boundary embedding is an injective continuous equivariant map ${\partial \Gamma\rightarrow F}$. It is antipodal (resp. generic) if pairs of distinct points are mapped to antipodal pairs (resp. pairs in general positions) in ${F}$.

Note that existence of a boundary embedding iplies that ${\rho}$ is discrete with finite kernel.

Definition 8 Say ${\rho}$ is Anosov if

1. it admits a boundary embedding ${\beta}$,
2. let ${r}$ be a geodesic ray in ${\Gamma}$ emanating from the identity element. Then there are uniform constants ${A}$ and ${c>0}$ such that

$\displaystyle \begin{array}{rcl} \epsilon(\rho(r(t))^{-1},\beta(r(\infty)))\geq Ae^{ct}, \end{array}$

where

$\displaystyle \begin{array}{rcl} \epsilon(g,x)=\min_{v\in T_x F,\,|v|=1}(Dg)_x(v). \end{array}$

We can prove that uniformity of constants is not crucial here.

4.2. Regularity

A symmetric space has a chamber valued distance, inspired by the theory of Coxeter groups. Indeed, ${K\setminus G/K=\Delta}$ is a Weyl chamber, i.e. a convex polygonal sector in ${{\mathbb R}^r}$.

Say a discrete subgroup ${\Gamma}$ of ${G}$ is regular if the distance of the chamber point ${d_\Delta(x,\gamma x)}$ to the boundary of the chamber tends to infinity as ${\gamma}$ tends to infinity in ${\Gamma}$. I.e. long segments joining orbit points tend to regular geodesics.

Say a discrete subgroup ${\Gamma}$ of ${G}$ is regular if the distance to the boundary is bounded below by an affine function of ${d(x,\gamma x)}$. This allows to define a visual limit set in the visual boundary which is in the regular part (the union of interiors of chambers in the Tits building structure).

4.3. Coarse geometry

Say a finitiely generated subgroup of ${G}$ is undistorted if orbit map is a quasi-isometric embedding.

Labourie and Guichard-Wienhard showed that Anosov representations are uniformly regular and undistorted. We prove the converse.

Theorem 9 ${\Gamma}$ hyperbolic, ${G}$ semi-simple. The following are equivalent:

1. ${\Gamma}$ is (non-uniformly) Anosov.
2. ${\Gamma}$ is uniformly regular and undistorted (URU).
3. ${\Gamma}$ is Morse.

Morse means that that long geodesic segments of ${\Gamma}$ are mapped into uniform neighborhoods of diamonds. A diamond is a segment in the sense of the chamber valued distance: the intersection of two Weyl chambers in the same maximal flat, pointing in opposite directions.

Morse property can be verified locally : on large enough balls.

4.4. Proof

Morse implies Anosov. A sequence of diamonds staying a bounded distance away from each other converge to a Weyl chamber.

Morse implies URU. Every ${\gamma}$ moves an apex ${x}$ along a maximal flat a definite amount, this implies that quasi-geodesics are mapped to quasi-geodesics.

Anosov implies Morse.

URU implies Morse. Use contraction properties of the projection to a maximal flat: uniformly regular quasi-geodesics are uniformly close to diamonds.

4.5. An application: construction of Schottky groups

Let ${\gamma_1}$ and ${\gamma_2}$ be axial geodesics, with regular axes and pair-wise generic end-points. Then, for large enough ${n_1}$, ${n_2}$, ${\gamma_1^{n_1}}$ and ${\gamma_2^{n_2}}$ generate a free, URU subgroup. This is a substitute for ping-pong, which does not seem to work easily in higher rank.

4.6. Question

Benoist uses a ping-pong argument somewhere. So do Breuillard and Gelander.

5. Bernhard Leeb: Geometry and dynamics of Anosov representations II

Joint work with Misha Kapovich.

I explain how the previous coarse considerations lead to study Finsler metrics on symmetric spaces, and compactifications that have good dynamical properties.

5.1. Finsler metrics

Pick a regular vector ${v}$ in the model Weyl chamber. This defines a linear function on the Weyl chamber, whence a Weyl group-invariant norm on the model maximal flat, whence a ${G}$-invariant metric on the symmetric space. It turns out that, given points ${p}$ and ${q}$, the diamond between them coincides with the union of all Finsler geodesics joining them.

5.2. Horoclosure

A proper geodesic metric space ${Y}$ embeds into continuous functions mod additive constants ${C(Y)/{\mathbb R}}$, via ${y\mapsto d(y,\cdot)}$. The closure ${\bar{Y}=Y\coprod \partial Y}$ of the image is compact. ${\partial Y}$ is called the horoboundary of ${Y}$.

If ${x_n\in Y}$ tends to infinity, to what does ${d(x_n,\cdot)}$ mod constant converge to ? If, seen from ${x}$, ${x_n}$ stays inside the Weyl chamber, the limit is a Busemann function which is linear, associated to ${v}$, in Weyl chambers. If ${x_n}$ approaches a singular ray, the limit is the max of two Busemann functions. In other words, compared to the visual (Tits) boundary,

• interiors of visual chambers are collapsed to points of the horoboundary.
• vertices of visual chambers are blown up into cells.

5.3. Properties of the horoboundary

Theorem 10 The ${G}$-equivariant compactification ${\bar{X}}$

1. is independent of the chosen Finsler metric,
2. has finitely many ${G}$-orbits corresponding to conjugacy classes of parabolic subgroups of ${G}$, or equivalently, to simplices of the Tits boundary.
3. has the structure of a manifold with corners,
4. is homeomorphic to a ball,
5. coincides with the maximal Satake compactification.

The largest (open) stratum is ${X}$, corresponds to the empty simplex. The smallest (closed) stratum is ${G/B}$, corresponding to maximal Weyl chambers. For intermediate simplices ${\tau}$, consider the space of strong asymptotic classes of Weyl sectors asymptotic to ${\tau}$.

Theorem 11 Let ${\Gamma}$ be a discrete subgroup of ${G}$.

1. There exist natural saturated domains ${\Omega=X\coprod \Omega_\infty \subset \bar{X}}$ such that ${\Gamma}$ acts properly discontinuously. This provides us with a bordification of ${X/\Gamma}$ as an orbifold with corners.
2. ${\Gamma}$ Anosov ${\Rightarrow}$ ${\Gamma}$ is cocompact on ${\Omega}$.

The converse holds.

Theorem 12 Let ${\Gamma}$ be a discrete subgroup of ${G}$. Assume that ${\Gamma}$ is unformly regular and acts cocompactly on ${\Omega}$, then ${\Gamma}$ is Anosov.

5.4. Proof

${\Omega}$ is obtained by removing a thickening ${Th(\Lambda)}$ of the limit set ${\Lambda}$.

5.5. An application to convergence actions

Let ${\Gamma}$ be a hyperbolic group, with a convergence action on a compact metrizable space ${\Sigma}$. Assume that the action on the limit set is conjugate to the action of ${\Gamma}$ on its ideal boundary. Assume that the complement ${\Omega}$ is path connected. Then ${\Gamma}$ is cocompact on ${\Omega}$.

This partially solves a question of Haissinsky.

6. Pierre-Emmanuel Caprace: Linear representations of lattices in Euclidean buildings

Joint work with Uri Bader and Jean Lecureux.

Keep in mind the following 3 pictures – a Euclidean plane tiled with equilateral triangles, – the incidence graph of the smallest projective plane, ${P^2(\mathbb{F}_2)}$, a bi-partite graph with 14 vertices. – a bi-colored tree.

6.1. Buildings and lattices

Here is a characterization of ${\tilde{A}_2}$-buildings.

Theorem 13 (Charney-Lytchak) An ${\tilde{A}_2}$-building is a simply connected 2-dimensional simplicial complex whose links are incidence graphs of projective planes.

Today, I will call ${\tilde{A}_2}$-lattice a pair ${(X,\Gamma)}$ where ${X}$ is a locally finite ${\tilde{A}_2}$-building and ${\Gamma}$ a discrete cocompact group of automorphisms of ${X}$.

Examples. ${X=}$ the Bruhat-Tits building of ${G=PGl(3,D)}$ where ${D}$ is a division algebra over a local field ${k}$ with finite residue field, ${\Gamma a cocompact arithmetic lattice (I call these the classical ${\tilde{A}_2}$-lattices).

Example 2 ${\Gamma_1=\langle x,y,z\,|\,x^7,y^7,z^7,xyz,x^3y^3z^3\rangle}$ is a cocompact lattice in ${G=PGl(3,\mathbb{F}_2((t)))}$.

There are non-classical ${\tilde{A}_2}$-buildings, I call them romantic.

6.2. Construction

Start with a finite 2-complex ${Y}$ satisfying the link condition. Then ${(\tilde{Y},\pi_1(Y))}$ is an ${\tilde{A}_2}$-lattice.

This can be implemented by a computer search. Non-trivial issue: determine wether ${\tilde{Y}}$ is classical or romantic.

Example 3 ${\Gamma_2=\langle x,y,z\,|\,x^7,y^7,z^7,xyz^3,x^3y^3z\rangle}$ is a romantic ${\tilde{A}_2}$-lattice.

It was discovered by Ronan and Tits in 1984, the presentation is due to Essert in 2011.

An enumeration of ${\tilde{A}_2}$-lattices with only one vertex was performed by Cartwright-Mantero-Steger-Zappa in 1994. In 1996, Barre found a new example. Essert continued Nicolas Radu recently beat the record, with an example where links are non-Desarguesian. Its thickness (degree of the link) is 10, which is the largest among known romantic examples. It is believed that romantic

6.3. Neo-classical examples ?

Let ${X}$ be the Bruhat-Tits building of ${G=PGl(3,D)}$. A theorem of Tits guarantees that

$\displaystyle \begin{array}{rcl} Aut(X)\equiv Aut(G)\equiv G\times Aut(k)\times \textrm{finite}. \end{array}$

A lattice ${\Gamma is called Galois if it has infinite image in ${Out(G)}$. Such lattices can exist only if ${\mathrm{char}(k)>0}$.

For instance, if ${k=\mathbb{F}_q((t))}$, ${Aut_{\mathbb{F}_q}(k)}$, known as the Nottingham group, is a huge pro-finite group (it contains copies of all pro-${p}$ groups).

Question. Does there exist a neo-classical lattice, i.e. a pair of a Bruhat-Tits building and a Galois lattice.

6.4. Properties

All ${\tilde{A}_2}$-lattices share the following properties:

1. ${X}$ is ${CAT(0)}$, ${\Gamma}$ is finitely presented.
2. ${\Gamma}$ has property (T) (Pansu-Zuk 1996).
3. ${\Gamma}$ is just-infinite (Shalom-Steger 2006).

6.5. A characterization of classical buildings

Theorem 14 Given an ${\tilde{A}_2}$-lattice, the following are equivalent:

1. ${\Gamma}$ has a linear representation with infinite image in ${Gl(d,F_0)}$, ${F_0}$ any field.
2. ${X}$ is the Bruhat-Tits building of ${G=PGl(3,D)}$ over ${k}$, ${\Gamma}$ is arithmetic, therefore virtually contained in ${G}$.

Therefore romantic or Galois lattices are not linear.

6.6. Reduction step

One can replace ${Gl(d,F_0)}$ with ${\mathbb{G}(F)}$ where ${\mathbb{G}}$ isa simple algebraic group over a local field ${F}$ and ${\rho(\Gamma)}$ is Zariski-dense. This uses Tits’ trick and property (T).

6.7. Transcendental step

Here, we use Bader-Furman’s Gate theory.

Let ${\Gamma}$ be a countable group, ${Y}$ a standard Borel space with a measure ${\nu}$, quasi-preserved by ${\Gamma}$. Gate theory associates to a linear reprentation ${\rho:\Gamma\rightarrow \mathbb{G}(F)}$ a continuous representation ${M\rightarrow\mathbb{H}(F)}$ for any polish group ${M}$ with an action on ${(Y,\nu)}$ commuting with ${\Gamma}$. ${\mathbb{H}}$ is a sub-quotient of ${\mathbb{G}}$.

Note that ${\Gamma}$ need not be contained in ${M}$. I explain what ${M}$ is in our setting. It is the projectivity group of the visual boundary.

6.8. Hilbert’s axiomatics for geometry

According to Hilbert 1897, a projective plane is a pair ${(P,L)}$ where ${L}$ (the set f lines) is a collection of subsets of the set ${P}$ of points, satisfying 3 axioms,

1. 2 distinct points lie on a unique line.
2. 2 distinct lines intersect in a unique point.
3. there exists 4 points, no 3 of which are collinear.

The classical, Desarguesian, examples are ${P^2(k)}$, ${k}$ a field or a skew field.

Hilbert discovered an infinite non-Desarguesian plane. A few years later, finite examples were found.

A projectivity is a permutation of a line ${L_0}$ produced by an alternating chain of points and lines.

Definition 15 Let ${(P,L)}$ be a projective plane. We fix a line and denote by ${M}$ its group of projectivities.

Fact. ${M}$ is 3-transitive.

For instance, if ${p=P^2(k)}$, then ${M=PGL(2,k)}$. For finite non-Desarguesian planes, ${M}$ is the full symmetric group.

6.9. ${M}$ on the boundary

In the same way as links in projective planes are lines, a building has a visual boundary which is a spherical building (appartments are 6-cycles). The idea of a projectivity generalizes. A closed chain of opposite vertices at infinity produces a permutation of the set of (singular) geodesics converging to the starting vertex. This set is a lower rank building, the group of permutations obtained is denoted by ${M}$.

For classical buildings, ${M}$ is the Levi part of a parabolic subgroup of ${G}$.

For ${\tilde{A}_2}$-buildings, a set of parallel singular geodesics is a regular tree ${T}$, ${M. We let ${M}$ and ${\Gamma}$ act on the space of geodesics, i.e. ${(Isom({\mathbb R},X)/{\mathbb R})}$. Let ${Y}$ be the closure of some orbit of ${M\times\Gamma}$. This turns out to admit an invariant measure, which is ergodic under ${\Gamma}$.

7. Damian Osajda: Gromov boundaries with the combinatorial Loewner property

Joint work with Antoine Clais.

7.1. Loewner property

Analysis on ideal boundaries of hyperbolic groups is powerful when ideal boundary has Loewner property.

Loewner property requires a ${Q}$-Ahlfors regular metric which has plenty of rectifiable curves. Plenty means that condensers ${(A,B)}$ (${A}$ and ${B}$ are disjoint continua) have finite and positive capacity, and that this capacity is of the order of

$\displaystyle \begin{array}{rcl} \frac{dist(A,B)}{\max\{\mathrm{diam}(A),\mathrm{diam}(B)\}}. \end{array}$

Loewner property turns out to be quasi-Möbius invariant only a posteriori.

Only a small list of examples have this property. Bonk and Kleiner introduced a variant which is genuinely quasi-Möbius invariant, and coined it the combinatorial Loewner property (CLP).

7.2. Combinatorial Loewner property

A ${\lambda}$-approximation of a compact metric space is a covering ${\Gamma_k}$ by open sets which contain disjoint balls of radius ${2^{-k}}$ and are contained in the ${\lambda}$-larger concentric balls.

Given a ${\lambda}$-approximation and positive function ${\rho}$ on ${\Gamma_k}$, a ${\rho}$-length is defined for curves by summing values of ${\rho}$ on pieces which intersect the curve. A ${(p,\rho)}$-mass is defined by summing ${\rho^p}$ over all pieces of the covering.

Given a family ${\mathcal{F}}$ of curves, minimizing ${(p,\rho)}$-mass over functions ${\rho}$ which give ${\rho}$-length ${\geq 1}$ to each curve of ${\mathcal{F}}$ yields the ${p}$-modulus ${M_p(\mathcal{F})}$.

Definition 16 (Bonk-Kleiner) Say a compact metric space ${Z}$ has the combinatorial Loewner property if for all pairs of disjoint continua ${A}$ and ${B}$, the family ${\mathcal{F}(A,B)}$ of curves joining ${A}$ to ${B}$ satisfies

$\displaystyle \begin{array}{rcl} Mod_p(\mathcal{F}(A,B))\sim \frac{dist(A,B)}{\max\{\mathrm{diam}(A),\mathrm{diam}(B)\}}. \end{array}$

Kleiner conjectures that if the ideal boundary of a hyperbolic group has the CLP, then its quasi-Möbius gauge contains a Loewner metric.

7.3. Examples

Examples of groups whose boundaries have CLP are still rare. I describe a new one. It is a right-angled Coxeter group.

The datum is finite graph, encoding pairs of involutions which are required to commute. The corresponding Coxeter group ${W}$ is hyperbolic iff the graph contains no cycles of length ${\leq 4}$.

Theorem 17 Consider the equilateral triangulation of the 2-torus whose combinatorial systole is ${4n}$, ${n\geq 2}$. Take its 1-skeleton. The corresponding Coxeter group has CLP.

From Antoine Clais’ work, it follows that ${W}$ is a lattice of automorphisms of the associated 3-dimensional hyperbolic building.

The ideal boundary ${\partial W}$ is homeomorphic to the Pontrijagin surface. Start with a 2-sphere, triangulate it, perform a connected sum with a 2-torus inside each face. triangulate again the obtained surface, perform connected sums, and so on, infinitely many times. It is a theorem of Jakobsche that the resulting topological space does not depend on choices. Fisher used it to describe ideal boundaries of cubical complexes

Indeed, the ${n+1}$-sphere in the building is obtained from the ${n}$-sphere by connect-summing 2-tori, one each time a vertex is passed.

The graph admits involutions which stabilize 2 cycles, one being pointwise fixed, the other not. We use pairs of such involutions, in order that fixed point sets disconnect the torus. This plays a role in the proof of CLP. Indeed, on ${\partial W}$, one needs to produce large families of curves following a given curve.

8. Anatoly Vershik: The absolute boundary of random walks on graphs and groups

On the classical subject of random walks on trees, I have a seemingly new result. Previously, I will explain a new conception of boundary, inspired by old papers by Dynkin. He used the words exit boundary and entrance boundary, which is a but improper, since there is no difference between exit and entrance. I prefer the word absolute.

8.1. Co-transition probabilities

Let ${G}$ be an oriented graph. For each vertex ${v}$, a probability distribution on outgoing edges is given. This equipment suffices to define the random walk.

One can also be given co-transition probabilities, i.e. conditional probabilities ${\Lambda}$ on incoming edges. The random walk is a measure on the space of paths, which detrmines co-transition probabilities, but the converse is not true.

Problem. Find all probability measures on the space ${T}$ of paths compatible with a given set ${\Lambda}$ of co-transition probabilities.

Definition 18 The absolute of ${(\Gamma,\Lambda)}$ is the list of all ergodic measures on ${T}$ compatible with ${\Lambda}$.

There is a symmetric (in fact identical) problem of determining all ergodic measures on ${T}$ compatible with a given set of transition probabilities.

Example 4 Usual random walk on ${{\mathbb Z}}$.

All paths through a point have the same co-transition probabilities (a power of ${1/2}$). So we must determine probability measures on ${T}$ giving uniform conditional measures (“central measure”). In other terms, a measure invariant under permutations of vertices. Only Bernoulli measures satisfy this. So the absolute is a interval ${[0,1]}$.

Our problem is expressible into operator algebra terms: characters of limits of finite…

Example 5 Discrete abelian groups.

I do not yet know the answer, but I am close to the solution.

8.2. Connection with Martin boundary

Absolute does not change if vertices are removed and their edges a replaced with direction connections. This operation changes Martin boundary.

Let ${\Sigma_n}$ denote the ${n}$-simplex. The map ${\pi_n:\Sigma_{n+1}\rightarrow\Sigma_n}$. The absolute is the set of extremal points of the inverse limit, which, according to Choquet, is again a simplex. It is contained in Martin boundary.

8.3. Random walks on trees

Consider simple uniform random walk on regular tree of degree ${q+1}$. The boundary is the set of ends.

Define the dynamic graph ${\mathcal{D}T_{q+1}}$ that records all paths (at level ${n+1}$, all vertices that can be reached after ${n}$ steps from the initial vertex).

We know that with probability 1, the trajectory reaches an end. This gives the ergodic decomposition of our set of trajectories. Equivalently, bounded harmonic functions correspond to functions on the space of ends. This is called Poisson boundary.

The conditional Markov process for which almost all trajectories reach a given point has the same co-transition probabilities. Does this provide all Markov processes in the absolute ?

Theorem 19

$\displaystyle \begin{array}{rcl} A(T_{q+1})=\partial T_{q+1}\times[\frac{1}{2},1]. \end{array}$

Let ${r\in[\frac{1}{2},1]}$, ${\omega\in\partial T_{q+1}}$. At each vertex, there is an edge pointing towards ${\omega}$. Put probability ${r}$ on this edge, and uniform probabilities on the other. This defines transition probabilities, hence a Markov process, compatible with given (uniform) co-transition probabilities. It is ergodic. ${r}$ can be interpreted as a drift, with speed ${\beta=2r-1}$. Only if ${r\geq 1/2}$ does the walk converge to ${\omega}$. For ${r=1/2}$, the speed equals to zero, i.e. distance to origin increases at sublinear speed (in fact, ${\sqrt{n}}$).

In the 1970’s, Stanislav Molchanov solved my problem in a different language, in terms of minimal non-negative eigenfunctions of the Laplacian. If ${\alpha}$ is an eigenvalue, the corresponding value of ${r}$ is

$\displaystyle \begin{array}{rcl} r_\alpha=\frac{1}{1+q^{1-2\alpha}}. \end{array}$

A phase transition occurs at ${1/2}$. If ${r<1/2}$, there is no ergodicity, trajectories are free to converge to any endpoint.

I begin having results for abelian groups, and some hope for nilpotent group. The picture is always the same: absolute fibres over Poisson boundary, with interval fibres.

9. Fanny Kassel: Proper affine actions for right-angled Coxeter groups

Joint work with Jeff Danciger and Francois Gueritaud.

There are still a number of open questions about affine manifolds. Auslander’s conjecture asserts that compact affine manifolds have solvable fundamental groups (partial answers: Fried-Goldman in dim 3, Abert-Goldman-Margulis in dim 4, under extra assumptions). At some point, it was not even clear wether compactness was necessary, until Margulis found free counterexamples in dimension 3.

Theorem 20 Every right-angled Coxeter group (RACG) with ${k}$ generators admits a proper action on ${{\mathbb R}^{k(k-1)/2}}$.

Many classes of groups embed in right-angled Coxeter groups: right-angled Artin groups (Davis-Januskiewicz), virtually special groups (Haglund-Wise) as well, this includes all Coxeter groups, surface groups, fundamental groups of hyperbolic 3-manifolds.

9.1. General setting

Let ${\Gamma}$ be a discret group. Let ${G}$ be a Lie group, with the action of ${G\times G}$. The infinitesimal version of this action is an affine action of ${G\times\mathfrak{g}}$ on the Lie algebra ${\mathfrak{g}}$. This suggests starting from pairs of representations of ${\Gamma}$, and move to pairs of a representation and a Lie algebra cocycle, since cocycles correspond to the tangent space of ${Hom(\Gamma,G)}$.

9.2. General principle: uniform contraction implies properness

2 years ago, we made the following observation. Start with a proper isometric action ${\rho}$ of ${\Gamma}$ on hyperbolic space ${\mathbb{H}^n}$. Let ${\rho':\Gamma\rightarrow G=O(n,1)}$ be another action which is uniformly contracting with respect to ${\rho}$. Then the pair ${(\rho,\rho'):\Gamma\rightarrow G\times G}$ is proper. There is an analogous statement for cocycles.

Uniformly contracting means that there exists strictly distance contracting equivariant map ${f:\mathbb{H}^n\rightarrow \mathbb{H}^n}$. The infinitesimal version deals with a pair of an action and a cocycle, and assumes existence of an equivariant vectorfield ${\mathbb{H}^n\rightarrow T\mathbb{H}^n}$ which contracts in the following sense: the flow contracts.

Here is a short proof. Map an isometry ${g}$ to the unique fixed point of ${g^{-1}\circ f}$. This is continuous and equivariant. Properness at target implies properness at source.

9.3. Other orthogonal groups

One needs all orthogonal groups ${O(p,q)}$ and their actions on spaces ${\mathbb{H}^{p,q}=P({\mathbb R}^{p,q+1})}$.

Theorem 21 Start with an isometric action ${\rho}$ of ${\Gamma}$ on space ${\mathbb{H}^{p,q}}$, preserving a proper convex domain ${\Omega}$ and proper on it. Let ${\rho':\Gamma\rightarrow G=O(n,1)}$ be another action which preserves a proper convex domain ${\Omega'}$ and which is uniformly contracting in space-like directions with respect to ${\rho}$. Then the pair ${(\rho,\rho'):\Gamma\rightarrow G\times G}$ is properly discontinuous.

There is an analogous statement for cocycles

The short argument does not generalize. Here is another proof which generalizes. I explain it in case ${(p,q)=(n,1)}$. Fix ${z\in \mathbb{H}^n}$ and stick to its orbit. Map ${g}$ to the set of points in ${\rho'(\Gamma)z}$ which minimize displacement by ${g^{-1}}$. It is a finite set. This equivariant map maps compact sets to compact sets, properness at target implies properness at source.

In the general case, convex sets are there to make sure that distances are used only in space-like directions.

9.4. Right-angled Coxeter groups

A finite graph, encoding commuting involutions, is given. The corresponding Gram matrix ${(-\cos(\pi/m_{ij}))=I-A}$ defines a symmetric bilinear form on ${{\mathbb R}^{k}}$, whence a canonical orthogonal representation of ${W}$ (Tits).

Let ${B_t=I-tA}$. This is nondegenerate of constant signature if ${t>t_0}$, whence representations ${\rho_t}$ in ${O(p,q+1)}$. In ${P({\mathbb R}^k)}$, the ${\Gamma}$-orbit ${\Omega_t}$ of the fundamental polyhedron ${P_t}$ is properly convex, and ${\Gamma}$ is cocompact on it (Tits-Vinberg). We check that ${\rho_t}$ is uniformly contracting with respect to ${\rho_{t'}}$ if ${t>t'}$. Also, the ${t}$-derivative satisfies uniform contraction in cocycle sense.

${\Omega_t}$ need not be contained in ${\mathbb{H}^{p,q}}$. If so, replace it with the intersection with the dual convex set, which is contained in ${\mathbb{H}^{p,q}}$, and non-empty. Examples show that contraction in non space-like directions does not hold.

10. Olivier Guichard: Symplectic maximal representations

Joint work with Anna Wienhard.

For such representations, which are Anosov, we construct domains of discontinuity in projective spaces, and would like to understand the topology of quotients.

10.1. Maximal representations of surface groups

${Sp(2n,{\mathbb R})}$ has a cyclic central extension ${\widetilde{Sp(2n,{\mathbb R})}}$, hence a characteristic class called the Euler number. On a surface subgroup ${\rho:\Gamma\rightarrow Sp(2n,{\mathbb R})}$ equipped with the standard presentation, the Euler number is given by

$\displaystyle \begin{array}{rcl} e=\prod_{i=1}^{g}[\widetilde{\rho(a_i)},\widetilde{\rho(b_i)}]\in{\mathbb Z}. \end{array}$

This readily implies the Milnor-Wood inequality

$\displaystyle \begin{array}{rcl} |e|\leq n(g-1). \end{array}$

Definition 22 Say ${\rho}$ is maximal if ${e(\rho)=n(g-1)}$.

Burger-Iozzi-Labourie-Wienhard show that maximal representations are Anosov. The boundary embedding ${\partial\Gamma\rightarrow\mathcal{L}}$,

$\displaystyle \begin{array}{rcl} \mathcal{L}=\{L\in Grassm(n,{\mathbb R}^{2n})\,;\,L\textrm{ is Lagrangian}\}, \end{array}$

is continuous, equivariant, antipodal (distinct points are mapped to transverse Lagrangians). It has a nice dynamical property: for every sequence ${(\gamma_n)}$ in ${\Gamma}$ such that there exist distinct boundary points ${t}$ and ${t'}$ such that, away from ${t}$, ${\gamma_n}$ converges to ${t}$, then ${\rho(\gamma_n)}$ converge to ${\beta(t)}$ away from the Lagrangians which are transverse to ${\beta(t')}$. This readily leads to proof of the proper discontinuity part of the following

Theorem 23 Define

$\displaystyle \begin{array}{rcl} K_\rho=\bigcup_{t\in\partial \Gamma}P(\beta(t))\subset P({\mathbb R}^{2n}). \end{array}$

If ${\rho}$ is Anosov, then the action of ${\Gamma}$ on the complement ${\Omega}$ of ${K_\rho}$ is properly discontinuous and cocompact.

10.2. Proof of cocompactness I: dynamics

Inspired by Sullivan, Kapovitch-Leeb-Porti. We use their expansion result. For every ${t\in\partial\Gamma}$, there exists ${C>1}$, a neighborhood ${U}$ of ${P(\beta(t))}$ in ${P({\mathbb R}^{2n})}$ such that for all ${t'\in\partial \Gamma}$ such that ${P(\beta(t'))}$ belongs to ${U}$, for all ${z\in U}$,

$\displaystyle \begin{array}{rcl} d(\rho(\gamma)z,\rho(\gamma)P(\beta(t')))\geq C\,d(z,P(\beta(t))). \end{array}$

This implies cocompactness.

10.3. Proof of cocompactness II: cohomology

Easier if ${n>2}$, since then ${\Omega}$ is connected. We show that compactly supported cohomology ${H^0_c(\Gamma\setminus\Omega)\not=0}$. Integrating along fibers, this is equal to ${H^2_c(\Gamma\setminus\Omega\times\tilde{\Sigma})\not=0}$, which fits in the long exact sequence of the pair

$\displaystyle \begin{array}{rcl} \end{array}$

So ${H^2_c(\Gamma\setminus\Omega\times\tilde{\Sigma})}$ is a cokernel whose dimension does not depend on ${n}$ and can be computed for ${n=1}$, it is equal to 1.

10.4. Proof of cocompactness III: maximality

A pair of transverse Lagrangians ${L^+}$ and ${L^-}$ defines a quadratic form on ${{\mathbb R}^{2n}}$.

Definition 24 A triple ${(L^+,L^0,L^-)}$ of pairwise transverse Lagrangians is maximal of the the restriction to ${L^0}$ of the quadratic form associated to

Theorem 25 (Burger-Iozzi-Wienhard) ${\rho}$ is maximal iff the boundary embedding ${\beta}$ is maximal, i.e. for every oriented triple ${(t^+,t^0,t^-)}$ in ${\partial\Gamma}$, ${(\beta(t^+),\beta(t^0),\beta(t^-))}$ is maximal.

${\beta}$ maps the oriented triple space of ${\partial\Gamma}$ to the maximal triple space of the Lagrangian Grassmannian. This space carries a ${O(n)}$ principal bundle ${\mathcal{E}}$, whose total space is the group ${Sp(2n,{\mathbb R})}$ itself. Let ${\mathcal{E}'}$ be the pulled-back bundle on triples${(\partial\Gamma)}$. One shows that the map at the level of total spaces is open. It is equivariant. The ${\Gamma}$-quotient ${\Gamma\setminus\mathcal{E}'}$ is compact, therefore the image is open and compact. If ${n>2}$, ${\Omega}$ is connected, therefore this map is onto, and ${\Gamma\setminus\mathcal{E}}$ is compact.

10.5. Topology of quotients

Theorem 26 The quotient manifold ${\rho(\Gamma)\setminus\Omega}$ fibres over ${\Sigma}$ with fibres homeomorphic to ${O(n)/\{\pm 1\}\times O(n-2)}$.

Since this statement is stable under deformation of the representation. We use knowledge of the connected components of the moduli space of maximal representations. Here is a list containing representatives of all connected components.

1. Fuchsian representations ${\Gamma\rightarrow Sl(2,{\mathbb R})}$.
2. Fuchsian composed with the irreducible representation ${\tau_{2n}:Sl(2,{\mathbb R})\rightarrow Sp(2n,{\mathbb R})}$.
3. ${n}$ Fuchsian representations composed with the fully reducible representation ${\tau_\pi:Sl(2,{\mathbb R})^n\rightarrow Sp(2n,{\mathbb R})}$.
4. A Fuchsian and an orthogonal representation composed with the tensor product representation ${\tau_\otimes:Sl(2,{\mathbb R})\times O(n)\rightarrow Sp(2n,{\mathbb R})}$.
5. Amalgamation: glue two surfaces with boundary ${\Sigma_\ell}$ and ${\Sigma_r}$ along their boundary. Then ${\pi_1(\Sigma)}$ is an amalgamation, compatible maximal representations define a representation of ${\pi_1(\Sigma)}$ which is still maximal (Burger-Iozzi-Wienhard).

11. Sourav Ghosh: Moduli space of Margulis space-times

Margulis space times are examples of complete affine 3-manifolds with free fundamental groups. Their holonomy is contained in ${SO_0(2,1)\times{\mathbb R}^3}$. The linear parts are discrete (Margulis started with certain Schottky groups).

In 1991, Drumm showed that one could start with an arbitrary Schottky group. He constructed fundamental domains bounded by what he called crooked planes. The fundamental domain of the Schottky group in hyperbolic plane is bounded by 4 lines. Add rays tangent to the isotropic circle. Consider the positive cone on this picture. Continue the picture to get complete half-planes. This bounds a fundamental domain.

Theorem 27 (Danciger-Gueritaud-Kassel) Any Margulis space time admits a fundamental domain bounded by crooked planes.

There are examples where the linear holonomy contains parabolics. From now on, I will stick to linear holonomies which are Schottky.

11.1. The neutralised section

The linear holonomy defines a hyperbolic surface ${\Sigma}$. In its unit tangent bundle, let ${U_{rec}\Sigma}$ denote the set of bi-recurrent points of the geodesic flow, and ${U_{rec}\mathbb{H}}$ its lift in the unit tangent bundle of hyperbolic plane. Similarly, there is a bi-recurrent set ${U_{rec}M}$ in the unit space-like tangent bundle of the affine 3-manifold ${M=\Gamma\setminus {\mathbb R}^3}$.

Theorem 28 (Goldman-Labourie-Margulis) There exists a continuous equivariant map ${N:U_{rec}\mathbb{H}\rightarrow{\mathbb R}^3}$, called the neutralised section, such that

$\displaystyle \begin{array}{rcl} N(\phi_t x)=N(x)+c(t)\nu(x). \end{array}$

Hence

$\displaystyle \begin{array}{rcl} \hat{N}=(N,\nu):U_{rec}\mathbb{H}\rightarrow U_{rec}{\mathbb R}^3 \end{array}$

is an orbit preserving homeomorphism.

However, horocycles are not mapped to horocycles. Each of them is mapped into a fixed affine plane.

Theorem 29 Let us define the “new horocycles in ${U_{rec}M}$” as …. Then these sets are stable leafs under the geodesic flow of a metric on ${M}$ which is bi-Lipschitz equivalent to a Euclidean metric.

11.2. Affine Anosov representation

Definition 30 Let ${\Gamma}$ be a free group, let ${W^\pm}$ be two transverse null affine planes (parallel to planes tangent to the light cone). Let ${P^\pm}$ be their stabilizers in the affine group ${G=SO_0(2,1)\times{\mathbb R}^3}$. We say that a representation of ${\Gamma}$ in ${G}$ is ${(G,P^\pm)}$-affine Anosov if there is a continuous equivariant boundary map on pairs of distinct points of ${\partial\Gamma}$ to ${G/P^+\cap P^- \subset G/P^+ \times G/P^-}$, and a continuous equivariant map from the space of the geodesic flow of ${\Gamma}$ to the bundle of Euclidean metrics on ${TG/P}$ showing contraction under the geodesic flow of ${\Gamma}$.

Theorem 31 An affine action of a Schottky subgroup of ${SO_0(2,1)}$ on ${{\mathbb R}^3}$ gives rise to a Margulis space time if and only if it is affine Anosov.

In higher dimensions, it is easier to establish the Anosov character of affine actions than to prove their properness.

12. Swiatoslav Gal: Simplicity of groups of dynamical origin

Certain simple groups are simpler than others.

Note that a group ${\Gamma}$ is simple iff for every nontrivial conjugacy class ${C}$, the union of its powers is the whole group.

Say ${\Gamma}$ is boundedly simple if for every nontrivial conjugacy class ${C}$, there exists ${N}$ such that

$\displaystyle \begin{array}{rcl} \Gamma=\bigcup_{n=-N}^N C^n. \end{array}$

Full topological groups. Matsui proved that the commutator group of a full topological group is simple. His proof shows that it is boundedly simple. The exponent ${N}$ indeed depends on the element.

Say ${\Gamma}$ is uniformly simple if there exists ${N}$ such that for every nontrivial conjugacy class ${C}$,

$\displaystyle \begin{array}{rcl} \Gamma=\bigcup_{n=-N}^N C^n. \end{array}$

12.1. Pseudo-lengths

A function ${\ell:\Gamma\rightarrow{\mathbb R}_+}$ is called a pseudo-length if ${\ell(e)=0}$ and for all ${g,h\in\Gamma}$,

$\displaystyle \begin{array}{rcl} \ell(gh)\leq\ell(g)+\ell(h). \end{array}$

Say ${\ell}$ is central if furthermore ${\ell(gh)=\ell(hg)}$.

I observe that ${\Gamma}$ is simple iff every central pseudo-length is a length. Also, ${\Gamma}$ is boundedly simple iff every central pseudo-length is a bounded length. For uniform simplicity, I have only one implication,

${\Gamma}$ uniformly simple ${\Rightarrow}$ every central pseudo-length is a bounded and discrete length.

12.2. Displacement and uniform simplicity

Burago-Ivanov-Polterovich discuss bi-invariant metrics on homeomorphism, diffeomorphism, symplectomorphism groups. They make the following observation: say that an element ${g}$ ${k}$-displaces a subgroup ${H}$ if successive conjugated subgroups ${H,{}^g H,\ldots,{}^{g^{k-1}}H}$ mutually commute. It this holds, then every element ${f}$ of the commutator subgroup ${H'}$ with commutator length ${\leq k}$ in fact ca be written as a prduct of two commutators,

$\displaystyle \begin{array}{rcl} f=[\alpha,\beta][\gamma,g]. \end{array}$

The trick is an identity among commutators.

A consequence: if ${g}$ has the property that every finitely generated subgroup is ${k}$-displaced by some conjugate of ${g}$, then ${\Gamma'\subset ({}^{\Gamma'}g)^6}$. In other words, ${\Gamma'}$ is uniformly 6-simple.

12.3. Application: actions on ordered sets

Let ${\Gamma}$ act on a totally ordered set. Assume that

1. action is bounded (the support of every group element is contained in some interval),
2. action is proximal (every interval is mapped into any other interval by some group element).

Then the the commutator subgroup ${\Gamma'}$ is uniformly 6-simple.

Note that bounded+proximal is equivalent to bounded+primitive (action is primitive if there are no factor ordered sets).

Example: Thompson’s group acting on the interval.

Example: affine action of ${{\mathbb Z}[1/p]}$ on ${{\mathbb R}}$ is proximal, not 2-transitive (which is an easy sufficient condition for proximality) but unbounded.

12.4. Groups of PL maps of the interval

Here are examples of groups of interest. Automorphism group of a tree. The corresponding full topological group (known as Neretin group). Group of planar tree automorphisms (for some planar embedding of the tree) and its full topological group.

Theorem 32 Equivalent properties for a group acting on the ideal boundary of a tree.

1. ${\Gamma}$ is proximal.
2. The full group ${\|\Gamma\|}$ is proximal.
3. ${\Gamma}$ action is minimal and does not preserve any probability measure.
4. ${\Gamma}$ action is minimal and non-parabolic.

Theorem 33 If ${\Gamma=\|\Gamma\|}$ and is proximal, then ${\|\Gamma\|}$ is uniformly simple.

12.5. More groups

The quasi-isometry group of a tree is uniformly simple as well (with Nir Lazarovitsch). The bound is better than in the previous class.

Volodymyr Nekrashevich has a paper where he shows that certain full topological groups are simple and finitely generated. He uses a different language.