Almost-automorphisms of trees, cloning systems and finiteness properties

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 $V_d(G)$.