Some isomorphism results for Thompson like groups $V_n(G)$

TL;DR

This paper investigates isomorphism conditions for Thompson-like groups $V_n(G)$, showing that when $G$ acts freely, these groups are isomorphic to $V_n$, and explores broader isomorphism patterns within this family.

Contribution

The paper proves that $V_n(G)$ is isomorphic to $V_n$ if and only if $G$ acts freely, and extends this to identify additional isomorphisms among these groups.

Findings

$V_n(G) \,\cong\, V_n$ when $G$ acts freely

Non-isomorphism results when $G$ does not act freely

Examples of further isomorphisms within the family $V_n(G)$

Abstract

We consider a class of groups $V_n(G)$ which are supergroups of the Higman-Thompson groups $V_n$. These groups fit in a framework of Elizabeth Scott for generating infinite virtually simple groups, and the groups we study in particular are initially introduced by Farley and Hughes. The group $V_n(G)$ is the result one obtains by taking the $V_n$ generators and adding a tree automorphism for each generator of a subgroup $G$ of the symmetric group on $n$ letters, where the new generators each permute the child leaves of a specific vertex $\alpha$ of the infinite rooted $n$-ary tree according to the permutation they represent, and then they iterate this permutation again at each vertex which is a descendent of $\alpha$. Farley and Hughes show that $V_n(G)$ is not isomorphic to $V_n$ when $G$ fails to act freely on the points $\{1,2,...,n\}$, and expect further non-isomorphism results in the other cases. We show the perhaps surprising result that if $G$ does act freely, then $V_n(G)\cong V_n$. We also generalise these results and produce some examples of even more isomorphisms amongst groups in the family $V_n(G)$. Essential tools in the above work are a study of the dynamics of the action of elements of $V_n(G)$ on Cantor space, Rubin's Theorem, and transducers from Grigorchuk, Nekrashevych, and Suschanski\u{i}'s rational group on the $n$-ary alphabet.