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

Abstract

Abstract. We introduce the notion of the k-closure of a group of automorphisms of a locally finite tree, and give several examples of the construction. We show that the k-closure satisfies a new property of automorphism groups of trees that generalises Tits' Property P . We prove that, apart from some degenerate cases, any non-discrete group acting on a tree with this property contains an abstractly simple subgroup.

Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras

“…, k}. Since also G 0 ≥ G (0,1) is a proper inclusion, we have the chain of proper inclusions G 0 G (0,1) = G 1 G (1,2) . This shows that G (k,k+1) ≤ G 0 is not a maximal subgroup.…”

“…After choice of a permutation group F ≤ S n , Burger-Mozes construct groups U(F ) and index two subgroups U(F ) + acting on the n-regular tree in such a way that their local action around vertices is prescribed by F . These groups U(F ) + attract particular interest of the totally disconnected group community, since they provide concrete examples of abstractly simple and compactly generated non-discrete groups [8,9,2,36]. Applying Theorem A and combining it with known type I results [1,10], we give a complete characterisation of type I groups in this important class of examples.…”

Groups acting on trees with almost prescribed local action

“…Definition 4.1. We say that a group G ≤ Aut(T ) satisfies the edge-independence property if ϕ e is an isomorphism for every choice of e. This means that for every edge e, the pointwise stabilizer of e in G acts independently on the two half-trees emanating from e. The edge-independence property already appeared in [Ama03, BEW15] (it is called "independence property" in [Ama03] and "property IP 1 " in [BEW15]), and is strictly weaker than Tits' independence property (P). However, one can check that these are equivalent for closed subgroups of Aut(T ) (see for instance [Ama03, Lemma 10]).…”

Chabauty Limits of Simple Groups Acting on Trees

“…where B(v, k) is the ball centered at v and of radius k in . That notion was first introduced and studied by Banks-Elder-Willis in [3], in the case where is a tree, even though they used the notation J (k) instead of k J .…”

“…An important tool in the proofs of the results above is provided by the notion of k-closures recently introduced by Banks-Elder-Willis [3], some of whose properties are reviewed in § 3 below. We establish a key relation between Chabauty convergence and k-closures in the general context of automorphism groups of locally finite graphs, see Proposition 3.2.…”