## Notes of Camille Horbez’ lecture

Horoboundary of Outer space, and growth under random automorphisms

1. Random growth

Question. Pick an element ${g}$ of free group ${F}$. Apply a sequence of random elements of ${Aut(F)}$. How fast does the length grow after cyclic reduction ?

Theorem 1 Let ${g\in F}$. Let ${\mu}$ be a probability measure on ${Out(F)}$ whose support is finite and generates ${Out(F)}$. Let ${(\Phi_n)}$ be the corresponding random walk on ${Out(F)}$. Then the limit

$\displaystyle \begin{array}{rcl} \lim |\Phi_n(g)|^{1/n}=\lambda>1 \end{array}$

exists almost always.

This is an analogue of Furstenberg’s theorem for ${Sl(N,{\mathbb Z})}$, and of Anders Karlsson for mapping class groups.

2. Oseledec type result

Here is a classical refinement of the above theorems.

Theorem 2 (Furstenberg-Kiefer, Hennion) There is a deterministic filtration ${L_i}$ of ${{\mathbb R}^N}$ and Lyapunov exponents ${\lambda_i}$

My version:

Theorem 3 There is a deterministic tree of subgroups ${H}$ in ${F=F_N}$ and Lyapunov exponents ${\lambda_H}$ such that the growth has rate ${\lambda_H}$ for elements of ${F}$ conjugated into node ${H}$ but in none of its children.

There are at most ${\frac{3N-2}{4}}$ different positive Lyapunov exponents.

3. Horoboundary

This classical tool (Gromov ?) is used in the proof. Let ${X}$ be a (possibly non symmetric) metric space. Map a point ${x\in X}$ to the distance function, up to an additive constant. This maps ${X}$ to ${C(X)/{\mathbb R}}$, equipped with the topology of uniform convergence on compact sets.

Proposition 4 (Walsh) Assume that ${X}$ is geodesic, proper. Then The embedding is a homeomorphism onto its image, whose closure is compact.

Example. Horoboundary of the real line has 2 points.

We apply the following general fact to Outer space.

Theorem 5 (Karlsson-Ledrappier) Asymptotically, the growth of the distance to the origin of a random walk is modelled on the growth of a (random) horofunction ${h}$. I.e., if ${(\Phi_n)}$ is a random walk on a discrete group acting isometrically on ${X}$,

$\displaystyle \begin{array}{rcl} \lim\frac{1}{n}d(x_0,\Phi_n^{-1}(x_0)=\lim\frac{-1}{n}h(\Phi_n^{-1}(x_0)). \end{array}$

4. Outer space

On Outer space (the space of free actions of ${F}$ on trees), we use the Francaviglia-Martino distance (Lipschitz distance). According to White, it is equal to the log of the supremal ratio of translation lengths. It is achieved by an element which is represented, on the quotient graph, by a simple loop, a figure 8 or a pair of glasses. In particular, it is a primitive element. This makes this distance handily computable.

The Ciller-Morgan compactification is obtained when mapping trees to their translation length, viewed as a function on ${F}$, up to rescaling. I modify this construction by restricting to primitive elements of ${F}$, getting what I call the primitive compactification. Elements in the closure are interpreted as isometric actions on ${{\mathbb R}}$-trees.

Theorem 6 The horocompactification of Outer space is homeomorphic to the primitive compactification. This in turn is a proper quotient of the Culler-Morgan compactification.

Example. If orbits of ${F}$ on the real tree ${T}$ are dense, then the equivalence class of ${T}$ is reduced to ${T}$. But this us not always the case. Some equivalence classes are indeed non trivial

## Notes of Dominik Gruber’s lecture

Acylindrical hyperbolicity of graphical small cancellation groups

With Sisto.

We prove the theorem in the title and use it to exhibit new behaviours for the divergence function of a group.

Graphical small cancellation

It is an extension of small cancellation theory, devised by Gromov, in order to construct a finitely presented group weakly containing an expander. Gromov’s full construction uses (pseudo-)random choices, so the resulting presentation is not explicit. We shall not need these unpleasant steps, our presentations will be explicit.

Data: a graph ${\Gamma}$, edge orientations, edge labels in ${S}$. Consider the set of words read along closed paths. This is a normal subgroup of a free group, hence a quotient group ${G(\Gamma)}$.

By construction, ${\Gamma}$ maps to ${Cay(G(\Gamma),S)}$. Need not be injective, unless we add assumptions: small cancellation.

A piece ${p}$ is a labelled path that has at least two dustinct label-preserving maps to ${\Gamma}$. Say ${\Gamma}$ (and ${G(\Gamma)}$) satisfies ${C'(\lambda)}$ if ratios length of pieces over girth of ${\Gamma}$ are ${\leq \Gamma}$.

Classical small cancellation amounts to ${\Gamma}$ being a union of cycles. The language of van Kampen diagrams applies here as usual.

Theorem 1 (Gromov, Ollivier 2006) If ${\Gamma}$ is a finite ${C'(1/6)}$ graph, then ${G(\Gamma)}$ is hyperbolic, and every component of ${\Gamma}$ embeds isometrically into ${Cay(G(\Gamma),S)}$.

Theorem 2 (Gruber 2012) Let ${\Gamma=\coprod_N \Gamma_n}$ is ${C'(1/6)}$ graph, then ${G(\Gamma)}$ is lacunary hyperbolic (i.e. at least one asymptotic cone is a real tree)

Theorem 3 (Gruber-Sisto) Let ${\Gamma=\coprod_N \Gamma_n}$ is ${C'(1/6)}$ graph, then ${G(\Gamma)}$ is acylindrically hyperbolic.

1. Acylindrical hyperbolicity

See Hume and Sisto’s talks. The hyperbolic space ${Y}$ on which ${G(\Gamma)}$ acts is the Cayley graph of ${G(\Gamma)}$ with respect to the (infinite) generating system consisting in ${S}$ and the set of all words read along paths in ${\Gamma}$.

We use Strebel’s classification of geodesic triangles in ${C'(1/2)}$ small cancellation groups. There are 7 cases, the hyperbolicity constant ${\delta}$ is at most 2. Strebel’s argument goes through and shows that ${Y}$ is 4-hyperbolic.

We show that all hyperbolic elements satisfy WPD. Thin quadrangles have width at most 2 in the middle.

2. Divergence

${Div(n)}$ measures the geodesic distance outside balls of radius ${\leq n/2}$. This a quasiisometry invariant. Examples with linear, quadratic, cubic, exponential divergence are known.

We show examples where the lim inf of ${Div(n)/n^2}$ is 0, but the lim sup of ${Div(n)/f(n)}$ is ${+\infty}$ for every prescribed subexponential function.

We use large powers a WPD element to produce bridges that reduce divergence. Since, at every finite step of the construction, the group is hyperbolic, and this has exponential divergence, we may keep adding larger and larger relators to produce large (but subexponential) values of divergence.

## Notes of Indira Chatterji’s Rennes lecture

${CAT(0)}$-cube complexes and the median class

Joint with Talia Fernos and Alessandra Iozzi.

1. Motivation

The following corollary.

Theorem 1 A cocompact, irreducible lattice in ${Sl(2,{\mathbb R})\times Sl(2,{\mathbb R})}$ is not cubical.

Completed by Fernos, Caprace, Lecureux in order to prove that anay such lattives, when acting isometrically on a ${CAT(0)}$ cube complex, must have a fixed point.

Consider lattices in semi-simple Lie groups ${G}$.

If ${G}$ has property (T), they can’t be cubical. If ${G=SO(3,1)}$, they are cubical (Bergeron-Wise, using Kahn-Markovic, unobvious). For ${SO(4,1)}$, some lattices are cubical, as Anne Giralt explained, but for the other ones we don’t know.

2. Main result

Theorem 2 Let ${\Gamma}$ act isometrically on a ${CAT(0)}$ cube complex ${X}$ in a non elementary manner (no fixed point on ${X}$ nor on the visual boundary ${\partial X}$). Then a certain bounded cohomology class ${m\in H_b^2(\Gamma,\pi)}$ vanishes.

Corollary 3 (Superrigidity) Let ${\Gamma}$ be a cocompact irreducible lattice in a product of locally compact groups ${G}$. Let ${\Gamma}$ act essentially and non-elementarily on a ${CAT(0)}$ cube complex. Then the action extends continuously to ${G}$, factoring via one of the factors.

The fact that this follows from the theorem is due to Shalom and Burger-Monod.

3. The median class

3.1. Case of trees

I explain the case when ${X}$ is a tree.

Let ${H}$ be the set of oriented paths of length 2 in the tree. Let ${\pi}$ be the obvious action of ${\Gamma}$ on ${\ell^2(H)}$. Let ${w:X\times X\rightarrow \ell^2(H)}$ be defined by

$\displaystyle \begin{array}{rcl} w(x,y)=1_{[[x,y]]}-1_{[[y,x]]}, \end{array}$

where ${[[x,y]]}$ denotes the set of ${a\in H}$ which are between ${x}$ and ${y}$. This is unbounded, but the coboundary ${dw}$ is bounded. Indeed, cancellations leave us only with paths that touch the median.

In the case of trees, this is not surprising (classical fact that generalizes to hyperbolic metric spaces).

3.2. Median metric spaces

In a metric, the side ${I(x,y)}$ is the set of points for wich the triangle inequality is an equality. A metric space is median if given 3 points, there is a unique common point to the 3 sides.

${CAT(0)}$ cube complexes equipped with the metric which is ${\ell^1}$ on cubes are median.

3.3. Case of CCC

${CAT(0)}$ cube complexes have half-spaces: start cutting s cube in equal parts and continue for ever in contiguous cubes. Say half-spaces ${h_1 \subset h_2}$ are tightly nested if any half-space that sits in between must be one of them. Define ${H}$ as the set of pairs of tightly nested half-spaces. The same formula defines a 1-cochain ${w}$. The same cancellations show that ${dw}$ only involves pairs touching the median point. Therefore it is bounded. However, the bound depends on the dimension of ${X}$. Pull-back ${dw}$ on ${\Gamma}$ via an orbit.

3.4. Non vanishing

Burger-Monod show that

$\displaystyle \begin{array}{rcl} H_b^2(\Gamma,\pi)\equiv ZL_{alt,*}^{\infty}(B,\pi)^{\Gamma}, \end{array}$

where ${B}$ is a Poisson boundary. We use the Roller compactification, defined as follows. ${X}$ embeds in the set of subsets of ${H}$ (a point is mapped to the set of half-space pairs that contain it). Take the closure of the image of that embedding. Then (Zimmer), there is an equivariant map of ${B}$ to the set of probability measures on ${\bar{X}}$. One shows that the image is in ${\partial X}$, this gives the image of ${dw}$ as a nonzero cocycle on ${B}$.

## Notes of Romain Tessera’s Rennes lecture nr 2

Our main concern: let ${X}$ be a bounded degree graph which does not coarsely embed into Hilbert space. Does this imply that ${X}$ weakly contains an expander ? I will show that the answer is no.

1. A step towards a positive answer

Theorem 1 Let ${(G_n,S_n)}$ be a sequence of finite Cayley graphs, ${|S_n|=k}$. The following are equivalent.

1. ${Cay(G_n,S_n)}$ does not coarsely embed in Hilbert space.
2. There are probability measures ${\mu_n}$ on ${G_n\times G_n}$ such that

1. ${\mu_n\{(g,g')\,;\,d(g,g')\geq r_n\}}$ is bounded away from 0.
2. Poincaré inequality holds,

$\displaystyle \begin{array}{rcl} \sum_{x,\,y\in G_n}|f(x)-f(y)|^2\mu_n(x,y)\leq C\frac{1}{|G_n|}\sum_{x\sim y\in G_n}|f(x)-f(y)|^2. \end{array}$

The proof is by Hahn-Banach.

Question (James Lee). Can we take for ${\mu_n}$ the normalized counting measure on ${A_n\times A_n}$ for some subset ${A_n\subset G_n}$ ?

This would put us very close to getting an expander in ${G_n}$. I will show that the answer is again no. The counterexample relies on relative property (T).

2. Relative property (T)

Definition 2 Let ${G}$ be a finitely generated group. Let ${H}$ be an infinite subgroup of ${G}$. We say that the pair ${(G,H)}$ has relative property (T) if, for every affine isometric action of ${G}$ on a Hilbert space, the orbits of ${H}$ are bounded.

There is again a formulation in terms of 1-cocycles: such cocycles should be bounded along ${H}$.

Proposition 3 Suppose that the pair ${(G,H)}$ has relative property (T). Let ${G_n}$ be a sequence of finite quotients of ${G}$. Then there is ${C>0}$ such that

$\displaystyle \begin{array}{rcl} \sum_{x,\,y\in G_n}|f(x)-f(y)|^2\mu_n(x,y)\leq C\frac{1}{|G_n|}\sum_{x\sim y\in G_n}|f(x)-f(y)|^2. \end{array}$

with ${\mu_n}$ the uniform measure on the set of pairs ${(x,xh)}$, ${h\in H}$.

Clearly, this implies non embeddability into Hilbert space, but for a substantially different reason than expansion.

3. A counterexample

From now on, joint work with Goulnara Arzhantseva.

3.1. First attempt

Use ${G={\mathbb Z}^2 \times Sl(2,{\mathbb Z})}$, ${H={\mathbb Z}^2}$. This is known to have relative property (T) (Kazhdan). Take ${G_n=({\mathbb Z}/n{\mathbb Z})^2 \times Sl(2,{\mathbb Z}/n{\mathbb Z})}$ ? Unfortunately, ${Cay(Sl(2,{\mathbb Z}/n{\mathbb Z}),S)}$ is an expander.

3.2. Successful attempt

Replace ${Sl(2,{\mathbb Z}/n{\mathbb Z})}$ with a sequence ${T_n}$ of finite groups mapping onto them, but which coarsely embed into Hilbert space.

Let ${n=2^k}$. Let ${J_k\subset Sl(2,{\mathbb Z}/n{\mathbb Z})}$ the index 3 subgroup generated by matrices

$\displaystyle \begin{array}{rcl} \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix},\quad \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix},\quad \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}. \end{array}$

It has order ${2^{3k-1}}$.

For an arbitrary group ${G}$, consider the following characteristic subgroups ${\Gamma_0(G)=G}$, ${\Gamma_{k+1}(G)=}$ subgroup of ${\Gamma_k(G)}$ generated by squares.

We define ${H_n=F_3/\Gamma_n(F_3)}$. One easily checks that ${H_n}$ is a finite 2-group.

Lemma 4 Let ${G}$ be a finite 3-generated group. If ${|G|=2^n}$, ${\Gamma_n(G)=1}$, therefore there is an epimorphism ${H_n\rightarrow G}$.

Corollary 5 There is an epimorphism ${H_{3k-1}\rightarrow J_k}$, mapping the standard generators to the 3 matrices above.

Theorem 6 (Arzhantseva-Guentner-Spakula 2012) The sequence ${(H_n)}$ embeds uniformly into Hilbert space.

The interest of these examples is that the embedding does not arise from Yu’s property (A). Locally, ${H_n}$ looks more and more like a tree. The embedding arises from a wall-space structure. The first one ${H_1=({\mathbb Z}/2{\mathbb Z})^3}$ is a cube, and this continues.

3.3. Absence of weakly embedded expanders

Remember that ${G_n}$ the semi-direct product ${({\mathbb Z}/2^k{\mathbb Z})\times H_{3k-1}}$, where each factor coarsely embeds in Hilbert space. This suffices to prove that ${G_n}$ does not weakly contain an expander. Indeed, if there was one, compose with projection to ${H_{3k-1}}$. A subset ${A_n}$ representing a positive proportion of points would be mapped to a point in ${H_{3k-1}}$. Thus this subset is mapped to ${{\mathbb Z}/2^k{\mathbb Z}}$, and therefore to Hilbert space again.

Lemma 7 Let ${X_n}$ be an expander, let ${A_n\subset X_n}$, ${|A_n|\geq c|X_n|}$. Then, for all 1-Lipschitz maps of ${A_n}$ to Hilbert space,

$\displaystyle \begin{array}{rcl} \frac{1}{|A_n|^2}\sum_{x,\,y\in A_n\times A_n}|f(x)-f(y)|^2 \end{array}$

stays bounded.

Indeed, use the Poincaré inequality in ${X_n}$. Given a function ${f:A_n\rightarrow\mathcal{H}}$, define ${\tilde{f}:X_n\rightarrow\mathcal{H}}$ by ${\tilde{f}=f}$ on ${A_n}$ and ${\tilde{f}(x)=f(z_x)}$ where ${z_x}$ is the point of ${A_n}$ closest to ${x}$. Then

$\displaystyle \begin{array}{rcl} \frac{1}{|A_n|^2}\sum_{x,\,y\in A_n\times A_n}|\tilde{f}(x)-\tilde{f}(y)|^2 &\leq&\frac{1}{|X_n|^2}\sum_{x,\,y\in X_n\times X_n}|\tilde{f}(x)-\tilde{f}(y)|^2\\ &\leq&C\frac{1}{|X_n|}\sum_{x\sim y\in X_n}|\tilde{f}(x)-\tilde{f}(y)|^2\\ &\leq&C\frac{1}{|X_n|}(\sum_{x\sim y\in A_n}|\tilde{f}(x)-\tilde{f}(y)|^2\\ &&+\sum_{x\sim y\in B_1}|\tilde{f}(x)-\tilde{f}(y)|^2+\sum_{x\sim y\in B_2}|\tilde{f}(x)-\tilde{f}(y)|^2\cdots), \end{array}$

where ${B_i}$ is the set of points at distance ${i}$ from ${A_n}$. Using the isoperimetric inequality in ${X_n}$, one shows that ${|B_i|}$ decays exponentially. This leads to the required upper bound.

## Notes of Yves Cornulier’s Rennes lecture nr 2

During the first lecture, we saw that, up to compact and cocompact groups, locally compact groups of polynomial growth can be reduced to simply connected Lie groups. Therefore we continue with a thorough study of these groups.

1. The lower central series

${G^{(1)}=G}$, ${G^{(i+1)}=[G,G^{(i)}]}$. For simply connected nilpotent Lie groups, the Lie algebra functor is an equivalence of categories. In fact, thanks to the Baker-Campbell-Hausdorff formula, the group can be viewed as a multiplication on the Lie algebra.

Pick a complement ${V_i}$: ${\mathfrak{g}^{(i)}=V_i \oplus\mathfrak{g}^{(i+1)}}$. Fix a Euclidean norm on each ${V_i}$, denote by ${V_i(R)}$ the ${R}$-ball in ${V_i}$. Get a Euclidean norm on ${\mathfrak{g}}$ and a left-invariant Riemannian metric on ${G}$.

Guivarc’h showed that that ${r}$-ball in ${G}$ is squeezed between boxes

$\displaystyle \begin{array}{rcl} K(R)=\bigoplus_i V_i(R^i) \end{array}$

of respective radii ${R=r/C}$ and ${R'=Cr}$. It follows that growth is polynomial of degree

$\displaystyle \begin{array}{rcl} \delta=\sum_{i}i\,\mathrm{dim}(V_i). \end{array}$

Examples

1. The standard filiform Lie algebra ${\mathfrak{f}_n}$ has a basis ${x_1,y_2,\ldots,y_n}$ with only nonzero brackets ${[x_1,y_i]=y_{i+1}}$ for ${i=2,\ldots,n-1}$. The Lie group ${F_n}$ is a semi-direct product ${{\mathbb R}\times{\mathbb R}[t]/(t^{n-1})}$, where the generator ${s}$ of ${{\mathbb R}}$ acts by ${sP(t)=(1+t)^s P(t)}$. One can take ${V_1=}$ and ${V_i=}$ for ${i\geq 2}$, ${\delta=1+\frac{n(n-1)}{2}}$.
2. The Heisenberg Lie algebra ${H_{2n+1}}$ has a basis ${x_1,\ldots,x_n,y_1,\ldots,y_n,z}$ with nonzero brackets ${[x_i,y_i]=z}$. ${V_1=}$, ${V_2=}$, ${\delta=2n+2}$.
3. Upper unipotent matrices. Here ${\mathrm{dim}(V_i)=\max(0,n-i)}$.
4. Free ${s}$-nilpotent Lie group on ${k}$ generators.
5. Polynomial vectorfields on the line. Generators are ${e_i=x^{i+1}\frac{\partial}{\partial x}}$, nonzero brackets are ${[e_i,e_j]=(i-j)e_{i+j}}$.

2. Carnot Lie algebras

Definition 1 A nilpotent Lie algebra ${\mathfrak{g}}$ is Carnot if it satisfies one of the equivalent properties

1. ${\mathfrak{g}}$ admits a Lie algebra grading ${\mathfrak{g}=\bigoplus_i \mathfrak{g}_i}$ such that ${\mathfrak{g}^{(i)}=\bigoplus_{j\geq i}g_i}$.
2. ${\mathfrak{g}}$ has a contracting automorphism inducing a homothety on ${\mathfrak{g}/\mathfrak{g}^{(2)}}$.
3. ${\mathfrak{g}}$ has a self-derivation inducing a identity on ${\mathfrak{g}/\mathfrak{g}^{(2)}}$.
4. ${\mathfrak{g}}$ is isomorphic to ${Car(\mathfrak{g})}$.
5. The corresponding Lie group admits a proper, geodesic, left-invariant distance with non-isometric similarities.

Here ${Car(\mathfrak{g})}$, the associated Carnot algebra, is the natural Lie algebra structure on ${\bigoplus_{i}\mathfrak{g}^{(i)}/\mathfrak{g}^{(i+1)}}$. It can be thought of as a first order approximation of ${\mathfrak{g}}$.

In the above list, all examples are Carnot but the last one, polynomial vectorfields. The associated Carnot algebra is filiform.

3. Quasiisometry classification

Question. Does quasiisometry imply isomorphism ?

Pansu 1989 : quasiisometric nilpotent Lie groups have isomorphic associated Carnot Lie algebras.

Shalom 2001 : quasiisometric nilpotent Lie groups have the same Betti numbers.

4. Questions

Expansion of volume growth ? Breuillard gave an upper bound of ${r^{\delta-2/3s}}$ on the second term.

## Notes of David Hume’s lecture

Acylindrically hyperbolic groups from Kac-Moody groups

Joint with Caprace.

Recall that a group ${H}$ is hyperbolically embedded in ${G}$, denoted by ${H\hookrightarrow_h G}$, if there exists an isometric action of ${G}$ on some hyperbolic metric space ${X}$ such that

1. ${H}$ is quasiconvex in ${X}$.
2. Tubular neighborhoods of orbits tend not to intersect much, i.e. for all ${g\notin H}$,

$\displaystyle \begin{array}{rcl} \mathrm{diameter}((H+R)\cap(gH+R))\leq K(R). \end{array}$

3. ${H}$ acts properly on ${X}$.

${G}$ is acylindrically hyperbolic if it contains a free (non cyclic) hyperbolically embedded subgroup

Examples.

1. Hyperbolic groups.
2. Relatively hyperbolic groups.
3. Mapping class groups. Use action on curve complex.
4. ${Out(F_n)}$. Use action on free factor complex.
5. Cremona group. Use a perturbation of the action on infinite dimensional hyperbolic space.

Note that if ${H\hookrightarrow_h G}$ where ${H}$ is vitually ${{\mathbb Z}}$, then ${G}$ is not simple. Thus this gives a proof that Cremona group is not simple.

1. Kac-Moody groups

1.1. Definition

These are built from copies of ${Sl(2,{\mathbb R})}$ folowing a combinatorial recipe encoded in a generalized Cartan matrix ${A\in M_n({\mathbb Z})}$ such that

1. ${a_{ii}=2}$,
2. ${a_{ij}\leq 0}$,
3. ${a_{ij}=0\Leftrightarrow a_{ji}=0}$.

Example. ${Sl(n,{\mathbb R})}$ can be obtained in this manner. The matrix determines a (linear) graph which tells how to perform amalgamations.

If ${T}$ is a finite tree, the resulting group ${G_T}$ is an amalgam. This makes a rather weird looking group.

1.2. BN pairs

Such groups come with a saturated twin BN pair ${(B_+,B_-,N)}$, where ${G}$ is generated by ${B_+}$ and ${N}$, as well as by ${B_-}$ and ${N}$. ${B_+\cap B_-=T}$ is maximal abelian, ${W=N/T}$ is the finitely generated Coxeter group associated with the graph. ${\Delta_+=G/B_+}$ and ${\Delta_-=G/B_-}$ are Tits buildings of type ${(W,S)}$. They are hyperbolic iff ${W}$ is hyperbolic. There is a Chevalley automorphism ${\sigma}$ which swaps the roles of the two buildings, and ${\sigma(t)=t^{-1}}$ on ${T}$.

The orthogonal form of ${G}$ is the set of fixed points of ${\sigma}$. The Iwasawa decomposition states that ${G=KB_+=KB_-}$.

2. Results

Theorem 1 Let ${G}$ be a real Kac-Moody group with orthogonal form ${K}$. Assume that ${W}$ is infinite, non virtually abelian, and not a non trivial direct product (almost always the case provided the matrix is complicated enough). Then there exists a virtually ${{\mathbb Z}}$ group ${H}$ which is hyperbolically embedded in ${G}$.

I do not know wether the theorem is still true over finite fields.

2.1. Intermediate results

Theorem 2 Let ${G}$ be a group acting isometrically on a geodesic metric space ${X}$. Let ${h\in G}$ generate subgroup ${H}$. Assume that

1. projection onto some orbit is strongly contracting,
2. ${h}$ is weakly properly discontinuous, i.e. for all ${D}$ and ${x\in X}$, there is ${M}$ such that only finitely many elements ${g}$ of ${G}$ satisfy ${d(x,gx)\leq Dd(h^Mx,gh^Mx)}$.

Then there is a subgroup ${H'}$ containing ${H}$ as a finite index subgroup and which is hyperbolically embedded in ${G}$.

We use the following result of Caprace and Fujiwara. If a group acts on a ${CAT(0)}$ building. If some element ${h\in G}$ fixes an apartment ${\mathcal{A}}$ and ${h_{|\mathcal{A}}}$ is rank 1, then ${}$ has a strongly contracting orbit.

Theorem 3 Let ${G}$ be a group acting isometrically on a ${CAT(0)}$ building ${X}$. Assume that chamber stabilizers are finite. Assume that some element ${h\in G}$ acts as a hyperbolic isometry on ${X}$ with regular axis (far from any wall). Then ${h}$ is weakly properly discontinuous.

2.2. Proof of Theorem

The orthogonal form ${K}$ satisfies all assumptions in previous theorems.

## Notes of Miklos Abert’s lecture

1. Spectral radius of a random walk

Let ${G}$ be a countable group. Start walking randomly on ${G}$ (for instance, using uniform measure on a symmetric generating set). The Markov operator is

$\displaystyle \begin{array}{rcl} M:\ell^2G\rightarrow\ell^2G,\quad (fM)(x)=\frac{1}{|S|}\sum_{s\in S}f(xs). \end{array}$

The norm of ${M}$ is called the spectral radius of the random walk,

Kesten’s theorem states that ${G}$ is amenable iff the spectral radius vanishes.

2. Spectral index of a subgroup

Let ${H}$ be an infinite index subgroup of ${G}$. The spectral index of ${H}$ in ${G}$ is the norm of the Markov operator acting on ${\ell^2(Schreier(G/H,S))}$. One can think of it as a codimension. If ${H}$ is amenable, it equals the r-spectral radius of ${G}$.

Let ${G}$ act on probability space ${(X,\mu)}$. One can make spatial averages over ${(X,\mu)}$. The spectral radius on ${L^2(X,\mu)}$ equals 1, so remove constant functions. The spectral radius on ${L_0^2(X,\mu)}$ is called the global spectral radius ${\rho_{glob}}$ of the random walk. The local spectral radius ${\rho_{loc}}$ (on ${\ell^2}$ of the orbit of ${x}$) is always less,

$\displaystyle \begin{array}{rcl} \rho_{loc}(x)\leq\rho_{glob}. \end{array}$

Definition 1 Say action is Ramanujan if ${\rho_{loc}(x)=\rho_{glob}}$.

Example. Take a tower of finite index subgroups of a free group which does not have property ${\tau}$.

Example. Bernoulli actions are Ramanujan. Indeed,

Theorem 2