Almost-automorphisms of trees, cloning systems and finiteness properties

Abstract

We prove that the group of almost-automorphisms of the infinite rooted regular [Formula: see text]-ary tree [Formula: see text] arises naturally as the Thompson-like group of a so-called [Formula: see text]-ary cloning system. A similar phenomenon occurs for any Röver–Nekrashevych group [Formula: see text], for [Formula: see text] a self-similar group. We use this framework to expand on work of Belk and Matucci, who proved that the Röver group, using the Grigorchuk group for [Formula: see text], is of type [Formula: see text]. Namely, we find some natural conditions on subgroups of [Formula: see text] to ensure that [Formula: see text] is of type [Formula: see text] and, in particular, we prove this for all [Formula: see text] in the infinite family of Šunić groups. We also prove that if [Formula: see text] is itself of type [Formula: see text], then so is [Formula: see text], and that every finitely generated virtually free group is self-similar, so in particular every finitely generated virtually free group [Formula: see text] yields a type [Formula: see text] Röver–Nekrashevych group [Formula: see text].