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 Röver-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 Röver 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 Šunić 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 $F_{\infty }$ Röver-Nekrashevych group $V_d(G)$.

arXiv comment44 pages, 11 figures. v2: accepted version, published in J. Topol. Anal

References in corpus (14)

• The Algebra of Strand Splitting. I. A Braided Version of Thompson's Group V
• L-presentations and branch groups
• On amenability of automata groups
• Conjugacy and Dynamics in Thompson's Groups
• The braided Thompson's groups are of type $F_{\infty }$
• Maximal subgroups of groups of intermediate growth
• Examples and counter-examples of log-symplectic manifolds
• The Brin-Thompson groups sV are of type F_\infty
• A Thompson group for the basilica
• Rearrangement groups of fractals
• Thompson groups for systems of groups, and their finiteness properties
• Finitely presented groups associated with expanding maps
• Local similarities and the Haagerup property
• A user's guide to cloning systems

Cited by in corpus (3)

• On the finiteness length of some soluble linear groups
• Braiding groups of automorphisms and almost-automorphisms of trees
• Finiteness properties for relatives of braided Higman--Thompson groups