Trees, contraction and rank one groups
joint with Tom De Medts, Yves de Cornulier, Nicolas Monod and Romain Tessera
Question. Given a group , determine wether 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
– is closed in compact open topology on (ie action is proper).
– is not compact (no global fixed point).
– is transitive on Ends (in particular, is non discrete).
Example 2 – , a local field.
– regular tree, .
– Intermediate examples, like groups with prescribed local action (Burger-Mozes). – Kac-Moody groups.
Only the first class is linear. For instance, contains infinite products of finite groups.
Theorem 1 Let be a closed, non compact and boundary transitive subgroup of . The following are equivalent:
1. Stabilizers of boundary points are metabelian.
2. There exists a local field such that is contained in and contains .
Note that field characteristic plays no role.
Definition 2 Let be locally compact, let be an element of . The contraction group of is
Also, we shall need the parabolic group of ,
Example 3 If ,, then unipotents, triangular.
If where is a homothety, then , .
If is the semi-direct product of the product of copies of a finite cyclic group by acting by translations on . Let be the shift. Then is a dense and proper subgroup of the infinite product.
So need not be closed.
Theorem 3 Let be locally compact, non trivial, unimodular, without compact minimal subgroups. Then the following are equivalent.
1. There is an such that the closed subgroup generated by and is cocompact in and is metabelian.
2. equals or for locally compact non discrete.
21: pick a regular diagonal element.
12 uses 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 in this language).
Theorem 2 has a generalization.
Theorem 4 Let be locally compact, non trivial, unimodular, without compact normal subgroups. Then the following are equivalent.
1. There is an in such that the closed subgroup generated by and is cocompact in , and is closed and torsion free.
2. is virtually a rank one simple algebraic group over of a finite extension of or .
Definition 5 A locally compact group is hyperbolic if it is compactly generated and word hyperbolic with respect to some compact generating set.
Note that action on is discontinuous, but this does not prevent a nice theory. For instance, if is a hyperbolic metric space with a cocompact isometry group, then is locally compact hyperbolic. Also, a Lie group is locally compact hyperbolic iff it is simple of rank 1. Also, triangular matrices form a l.c. hyperbolic group.
Proposition 6 Let . Then the closed subgroup generated by and is always hyperbolic.
Indeed, put in a compact neighborhood of 1, get compact generating set . It generates , which is thus open and closed.
Now the closure of is exponentially distorted in (by Baire, there is such that is contained in . Therefore is contained in , this provides shortcuts. It follows that every geodesic of spends only a bounded amount of time in .
Corollary 7 If is locally compact and contains a such that B is cocompact, then is hyperbolic.
Crucial step: is amenable.
Theorem 8 G locally compact unimodular without nontrivial compact normal subgroups. The following are equivalent.
1. There is an such that the closed subgroup generated by and is cocompact in .
2. 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.