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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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