The Liouville property for groups acting on rooted trees (original) (raw)

Ergodic decomposition of group actions on rooted trees

2015

We prove a general result about the decomposition on ergodic components of group actions on boundaries of spherically homogeneous rooted trees. Namely, we identify the space of ergodic components with the boundary of the orbit tree associated with the action, and show that the canonical system of ergodic invariant probability measures coincides with the system of uniform measures on the boundaries of minimal invariant subtrees of the tree. A special attention is given to the case of groups generated by finite automata. Few examples, including the lamplighter group, Sushchansky group, and the, so called, Universal group are considered in order to demonstrate applications of the theorem.

Amenability of linear-activity automaton groups

Journal of the European Mathematical Society, 2013

We prove that every linear-activity automaton group is amenable. The proof is based on showing that a random walk on a specially constructed degree 1 automaton group-the mother group-has asymptotic entropy 0. Our result answers an open question by Nekrashevych in the Kourovka notebook, and gives a partial answer to a question of Sidki.

On the amenability and Kunze–Stein property for groups acting on a tree

1988

We characterize the amenable groups acting on a locally finite tree. In particular if the tree is homogeneous and the group G acts transitively on the vertices then we prove that G is amenable iff G fixes one point of the boundary of the tree. Moreover we prove that a group G which acts transitively on the vertices and on an open subset of the boundary is either amenable or a Kunze-Stein group. 1. Introduction and notations. Let X b e a locally finite tree, that is, a connected graph without circuits such that every vertex belongs to a finite set of edges. Let V be the set of vertices and E the set of edges. If V \ and v2 are in V, let [^1,^2] be the unique geodesic connecting v \ to v2; the distance d(v\, v2) is defined as the length of

Compact groups acting on trees

1980

In this note we shall extend a theorem of Bass on the action of profinite groups on trees to the case of an arbitrary compact topological group. Serre has introduced the notion of a group G of type FA in [5] ß this property FA guarantees that whenever G acts without inversions on a tree there is a fixed point. Bass [1 ] has investigated a weaker property FA', which insures a fixed point for every group element whenever G acts without inversions on a tree; this property FA' is equivalent to the group theoretic conditions (1)G has no infinite cyclic quotient and (2)G is not a non-trivial free product with amalgamation. We show that compact hausdorff groups have property FA'.

A zero-one law for random subgroups of some totally disconnected groups

Transformation Groups, 2009

Let A be a locally compact group topologically generated by d elements and let k > d. Consider the action, by precomposition, of Γ = Aut(F k) on the set of marked, k-generated, dense subgroups D k,A := {η ∈ Hom(F k , A) | φ(F k) = A}. We prove the ergodicity of this action for the following two families of simple, totally disconnected locally compact groups: • A = PSL 2 (K) where K is a non-Archimedean local field (of characteristic = 2), • A = Aut 0 (T q+1)-the group of orientation preserving automorphisms of a q + 1 regular tree, for q ≥ 2. In contrast, a recent result of Minsky's shows that the same action is not ergodic when A = PSL 2 (R) or A = PSL 2 (C). Therefore if K is a local field (with char(K) = 2) the action of Aut(F k) on D k,PSL2(K) is ergodic, for every k > 2, if and only if K is non-Archimedean. Ergodicity implies that every "measurable property" either holds or fails to hold for almost every k-generated dense subgroup of A.

Locally compact groups acting on trees and propertyT

Monatshefte f�r Mathematik, 1982

Following KAZHDAN, a separable locally compact group G is said to have property T if the trivial representation is isolated in the dual space, G, of equivalence classes of continuous irreducible unitary representations of G. We generalize results of MARGULIS--TITs by showing that groups which have property T can not be amalgams.

Self-similar groups acting essentially freely on the boundary of the binary rooted tree

2012

We study the class of groups generated by automata that act essentially freely on the boundary of a rooted tree. In the process we establish and discuss some general tools for determining if a group belongs to this class, and explore the connections of this class to the classes of just-infinite and scale-invariant groups. Our main application is a complete classification of groups generated by 3-state automata over 2-letter alphabet that are in this class.

Convergence to ends for random walks on the automorphism group of a tree

Proceedings of the American Mathematical Society, 1989

Let μ \mu be a probability on a free group Γ \Gamma of rank r ≥ 2 r \geq 2 . Assume that Supp ⁡ ( μ ) \operatorname {Supp} \left ( \mu \right ) is not contained in a cyclic subgroup of Γ \Gamma . We show that if ( X n ) n ≥ 0 {\left ( {{X_n}} \right )_{n \geq 0}} is the right random walk on Γ \Gamma determined by μ \mu , then with probability 1, X n {X_n} converges (in the natural sense) to an infinite reduced word. The space Ω \Omega of infinite reduced words carries a unique probability ν \nu such that ( Ω , ν ) \left ( {\Omega ,\nu } \right ) is a frontier of ( Γ , μ ) \left ( {\Gamma ,\mu } \right ) in the sense of Furstenberg [10]. This result extends to the right random walk ( X n ) \left ( {{X_n}} \right ) determined by a probability μ \mu on the group G G of automorphisms of an arbitrary infinite locally finite tree T T . Assuming that Supp ⁡ ( μ ) \operatorname {Supp} \left ( \mu \right ) is not contained in any amenable closed subgroup of G G , then with probability 1 ther...

Locally compact groups acting on trees

Pacific Journal of Mathematics, 1982

Following Serre's original description of groups having the fixed point property for actions on trees, Bass has introduced the notion of a group of type FA'.