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

The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms

Israel Journal of Mathematics, 2007

Given a free group F k of rank k ≥ 2 with a fixed set of free generators we associate to any homomorphism φ from F k to a group G with a left-invariant semi-norm a generic stretching factor, λ(φ), which is a noncommutative generalization of the translation number. We concentrate on the situation when φ : F k → Aut(X) corresponds to a free action of F k on a simplicial tree X, in particular, when φ corresponds to the action of F k on its Cayley graph via an automorphism of F k . In this case we are able to obtain some detailed "arithmetic" information about the possible values of λ = λ(φ).

The ergodic theory of lattice subgroups

2006

We prove mean and pointwise ergodic theorems for general families of averages on a semisimple algebraic (or S-algebraic) group G, together with an explicit rate of convergence when the action has a spectral gap. Given any lattice in G, we use the ergodic theorems for G to solve the lattice point counting problem for general domains in G, and prove mean and pointwise ergodic theorems for arbitrary measure-preserving actions of the lattice, together with explicit rates of convergence when a spectral gap is present. We also prove an equidistribution theorem in arbitrary isometric actions of the lattice. For the proof we develop a general method to derive ergodic theorems for actions of a locally compact group G, and of a lattice subgroup Gamma, provided certain natural spectral, geometric and regularity conditions are satisfied by the group G, the lattice Gamma, and the domains where the averages are supported. In particular, we establish the general principle that under these conditio...

Characteristic random subgroups of geometric groups and free abelian groups of infinite rank

2014

We show that if G is a non-elementary word hyperbolic group, mapping class group of a hyperbolic surface or the outer automorphism group of a nonabelian free group then G has 2^_0 many continuous ergodic invariant random subgroups. If G is a nonabelian free group then G has 2^_0 many continuous G-ergodic characteristic random subgroups. We also provide a complete classification of characteristic random subgroups of free abelian groups of countably infinite rank and elementary p-groups of countably infinite rank.

The structure of skew products with ergodic group automorphisms

Israel Journal of Mathematics, 1977

We prove that ergodic automorphisms of compact groups are Bernoulli shifts, and that skew products with such automorphisms are isomorphic to direct products. We give a simple geometric demonstration of Yuzvinskii's basic result in the calculation of entropy for group automorphisms, and show that the set of possible values for entropy is one of two alternatives, depending on the answer to an open problem in algebraic number theory. We also classify those algebraic factors of a group automorphism that are complemented.

2 Some Properties of Distal Actions on Locally Compact Groups

2016

We consider the actions of (semi)groups on a locally compact group by automorphisms. We show the equivalence of distality and pointwise distality for the actions of a certain class of groups. We also show that a compactly generated locally compact group of polynomial growth has a compact normal subgroup K such that G/K is distal and the conjugacy action of G on K is ergodic; moreover, if G itself is (pointwise) distal then G is Lie projective. We prove a decomposition theorem for contraction groups of an automorphism under certain conditions. We give a necessary and sufficient condition for distality of an automorphism in terms of its contraction group. We compare classes of (pointwise) distal groups and groups whose closed subgroups are unimodular. In particular, we study relations between distality, unimodularity and contraction subgroups.

On groups with Property (T_lp)

arXiv: Group Theory, 2013

Let 1 < p < ∞, p = 2. Property (T ℓp) for a second countable locally compact group G is a weak version of Kazhdan Property (T), defined in terms of the orthogonal representations of G on ℓ p. Property (T ℓp) is characterized by an isolation property of the trivial representation from the monomial unitary representations of G associated to open subgroups. Connected groups with Property (T ℓp) are the connected groups with a compact abelianization. In the case of a totally disconnected group, isolation of the trivial representation from the quasi-regular representations associated to open subgroups suffices to characterize Property (T ℓp). Groups with Property (T ℓp) share some important properties with Kazhdan groups (compact generation, compact abelianization, ...). Simple algebraic groups over non-archimedean local fields as well as automorphism groups of k-regular trees for k ≥ 3 have Property (T ℓp). In the case of discrete groups, Property (T ℓp) implies Lubotzky's Property (τ) and is implied by Property (F) of Glasner and Monod. We show that an irreducible lattice Γ in a product G 1 ×G 2 of locally compact groups have Property (T ℓp), whenever G 1 has Property (T) and G 2 is connected and minimally almost periodic. Such a lattice does not have Property (T) if G 2 does not have Property (T).

Automorphism groups of trees: generalities and prescribed local actions

New Directions in Locally Compact Groups

This article is an expanded version of the talks given by the authors at the Arbeitsgemeinschaft "Totally Disconnected Groups", held at Oberwolfach in October 2014. We recall the basic theory of automorphisms of trees and Tits' simplicity theorem, and present two constructions of tree groups via local actions with their basic properties: the universal group associated to a finite permutation group by M. Burger and S. Mozes, and the k-closures of a given group by C. Banks, M. Elder and G. Willis.

The structure of random automorphisms of the rational numbers

Fundamenta Mathematicae, 2020

In order to understand the structure of the "typical" element of an automorphism group, one has to study how large the conjugacy classes of the group are. For the case when typical is meant in the sense of Baire category, Truss proved that there is a co-meagre conjugacy class in Aut(Q, <), the automorphism group of the rational numbers. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. We give a complete description of the size of the conjugacy classes of Aut(Q, <) with respect to this notion. In particular, we show that there exist continuum many non-Haar null conjugacy classes, illustrating that the random behaviour is quite different from the typical one in the sense of Baire category.