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

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 $α$ 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 $α$. 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 Suschanskiĭ's rational group on the $n$-ary alphabet.

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