Almost-automorphisms of trees, cloning systems and finiteness properties

TL;DR

This paper explores the structure of groups related to automorphisms of infinite regular trees, establishing conditions under which these groups have strong finiteness properties, and extending known results to broader classes of groups.

Contribution

It introduces a framework connecting almost-automorphisms of trees with cloning systems, and proves new finiteness properties for R"over-Nekrashevych groups based on self-similar groups.

Findings

Groups of almost-automorphisms arise from $d$-ary cloning systems.

Conditions are identified for $V_d(G)$ to be of type $F_$.

Finitely generated virtually free groups produce $F_$ groups.

Abstract

We prove that the group of almost-automorphisms of the infinite rooted regular $d$-ary tree $\mathcal{T}_d$ arises naturally as the Thompson-like group of a so called $d$-ary cloning system. A similar phenomenon occurs for any R\"over-Nekrashevych group $V_d(G)$, for $G\le Aut(\mathcal{T}_d)$ a self-similar group. We use this framework to expand on work of Belk and Matucci, who proved that the R\"over group, using the Grigorchuk group for $G$, is of type $F_\infty$. Namely, we find some natural conditions on subgroups of $G$ to ensure that $V_d(G)$ is of type $F_\infty$, and in particular we prove this for all $G$ in the infinite family of \v{S}uni\'c groups. We also prove that if $G$ is itself of type $F_\infty$ then so is $V_d(G)$, and that every finitely generated virtually free group is self-similar, so in particular every finitely generated virtually free group $G$ yields a type $F_\infty$ R\"over-Nekrashevych group $V_d(G)$.