## Notes of Pierre-Emmanuel Caprace’s lecture

Trees, contraction and rank one groups

joint with Tom De Medts, Yves de Cornulier, Nicolas Monod and Romain Tessera

Question. Given a group ${G}$, determine wether ${G}$ has finite dimensional linear representations with infinite image.

Example 1 This was the main step of Gromov’s polynomial growth theorem. He used Hilbert’s 5th problem, a characterization of Lie groups among locally compact groups.

Problem: Characterize linear groups among locally compact groups.

Most of what is known deals with characteristic 0.

From now on, I focus on groups acting on trees. I assume

${G}$ is closed in compact open topology on ${Aut(T)}$ (ie action is proper).

${G}$ is not compact (no global fixed point).

${G}$ is transitive on Ends${(T)}$ (in particular, ${G}$ is non discrete).

Example 2 ${SL(2,K)}$, ${K}$ a local field.

${T}$ regular tree, ${G=Aut(T)}$.

– Intermediate examples, like groups with prescribed local action (Burger-Mozes). – Kac-Moody groups.

Only the first class is linear. For instance, ${Aut(T)}$ contains infinite products of finite groups.

Theorem 1 Let ${G}$ be a closed, non compact and boundary transitive subgroup of ${Aut(T)}$. The following are equivalent:

1. Stabilizers of boundary points are metabelian.

2. There exists a local field ${k}$ such that ${G}$ is contained in ${PGl(2,k)}$ and contains ${PSl(2,k)}$.

Note that field characteristic plays no role.

Definition 2 Let ${G}$ be locally compact, let ${a}$ be an element of ${G}$. The contraction group of ${a}$ is

$\displaystyle \begin{array}{rcl} U_a =\{g \in G \,;\, a^n ga^{-n}\textrm{ tends to }1\}. \end{array}$

Also, we shall need the parabolic group of ${a}$,

$\displaystyle \begin{array}{rcl} P_a =\{g \in G \,;\, a^n gg^{-n}\textrm{ is bounded}\}. \end{array}$

Example 3 If ${G=Sl(2,R)}$,${a=diag(2,1/2)}$, then ${U_a=\{}$unipotents${\}}$, ${P_a=\{}$triangular${\}}$.

If ${G=\mathbb{R}^n \times_a\mathbb{R}}$ where ${a}$ is a homothety, then ${U_a =\mathbb{R}^n}$, ${P_a = G}$.

If ${G}$ is the semi-direct product of the product of ${{\mathbb Z}}$ copies of a finite cyclic group by ${{\mathbb Z}}$ acting by translations on ${{\mathbb Z}}$. Let ${a}$ be the shift. Then ${U_a}$ is a dense and proper subgroup of the infinite product.

So ${U_a}$ need not be closed.

Theorem 3 Let ${G}$ be locally compact, non trivial, unimodular, without compact minimal subgroups. Then the following are equivalent.

1. There is an ${a \in G}$ such that the closed subgroup generated by ${a}$ and ${U_a}$ is cocompact in ${G}$ and ${P_a}$ is metabelian.

2. ${G}$ equals ${PGl(2,k)}$ or ${PSl(2,k)}$ for ${k}$ locally compact non discrete.

Proof:

2${\Rightarrow}$1: pick ${a}$ a regular diagonal element.

1${\Rightarrow}$2 uses ${U_a}$ to reduce to groups acting on trees, and then some structure theory of locally compact groups, and the theory of Moufang sets (there is a characterization of ${Sl(2,k)}$ in this language).

Theorem 2 has a generalization.

Theorem 4 Let ${G}$ be locally compact, non trivial, unimodular, without compact normal subgroups. Then the following are equivalent.

1. There is an ${a}$ in ${G}$ such that the closed subgroup generated by ${a}$ and ${U_a}$ is cocompact in ${G}$, and ${U_a}$ is closed and torsion free.

2. ${G}$ is virtually a rank one simple algebraic group over of a finite extension of ${\mathbb{Q}_p}$ or ${\mathbb{R}}$.

Definition 5 A locally compact group ${G}$ is hyperbolic if it is compactly generated and word hyperbolic with respect to some compact generating set.

Note that ${G}$ action on ${Cay(G,Q)}$ is discontinuous, but this does not prevent a nice theory. For instance, if ${X}$ is a hyperbolic metric space with a cocompact isometry group, then ${Isom(X)}$ is locally compact hyperbolic. Also, a Lie group is locally compact hyperbolic iff it is simple of rank 1. Also, triangular ${2\times 2}$ matrices form a l.c. hyperbolic group.

Proposition 6 Let ${a \in G}$. Then the closed subgroup ${B}$ generated by ${a}$ and ${U_a}$ is always hyperbolic.

Indeed, put ${a}$ in a compact neighborhood of 1, get compact generating set ${Q}$. It generates ${B}$, which is thus open and closed.

Now the closure ${C}$ of ${U_a}$ is exponentially distorted in ${G}$ (by Baire, there is ${N}$ such that ${a^N S^2 a^{-N}}$ is contained in ${S}$. Therefore ${a^{mN} S^{2m} a^{-mN}}$ is contained in ${S}$, this provides shortcuts. It follows that every geodesic of ${Cay(B,Q)}$ spends only a bounded amount of time in ${C}$.

Corollary 7 If ${G}$ is locally compact and contains a such that B is cocompact, then ${G}$ is hyperbolic.

Crucial step: ${B}$ is amenable.

Theorem 8 G locally compact unimodular without nontrivial compact normal subgroups. The following are equivalent.

1. There is an ${a \in G}$ such that the closed subgroup generated by ${a}$ and ${U_a}$ is cocompact in ${G}$.

2. ${G}$ is either a rank 1 simple Lie group or a closed subgroup of automorphisms of a tree, transitive on boundary.

This uses Hilbert’s 5th problem and Nevo’s PhD.