Generic groups acting on regular trees

Abstract

Abstract. Let T be a k-regular tree (k ≥ 3) and A = Aut(T ) its automorphism group. We analyze a generic finitely generated subgroup Γ of A. We show that Γ is free and establish a trichotomy on the closure Γ of Γ in A. It turns out that Γ is either discrete, compact or has index at most 2 in A.

Simple groups of automorphisms of trees determined by their actions on finite subtrees

“…Then inductively define x ∈ BS(m, n) (1) to agree with a i on B( a , 1), and to agree with a cv on B(v a , 1), where (u a ,v a ) is an edge, l is the smallest integer such that a l fixes the word v, and c v = c u + lσ v . These conditions ensure that a cv and a cu agree on the edge (u a ,v a ) and hence that x is an automorphism of T BS(m,n) , which is uniquely identified by the collection {σ v : v ∈ V (T BS(m,n) )}.…”

“…The main result of [1] implies that there exist free groups acting on regular trees which are dense in either Aut(T ) or the simple subgroup Aut(T ) + (see Section 7). Such a group cannot have Property IP 1 but its closure does.…”

Compactifications and algebraic completions of limit groups

“…Conversely, if axis(a)∩axis(b) is finite, for n > |axis(a)∩axis(b)|, a n and b n satisfy the Schottky condition (the condition of the ping-pong lemma, see [18,Chapter II.B. ] or [1]) and hence a n , b n ∼ = F 2 .…”

“…Little is known about the general theory of dense subgroups. Yet, given a dense or nondiscrete embedding of an abstract group Γ in some topological group G, one often deduces interesting information about Γ or G (see [16,14,22,28,1,4,5,13]). There are several standard methods (probabilistic, algebraic and dynamical) to produce dense free subgroups in a given G (see [11,4,5]).…”

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

“…Theorem ( [AG09a]). Let T = T q+1 be the (q + 1)-regular tree for q 2 and A = Aut 0 (T ) its automorphism group.…”