# Some isomorphism results for Thompson-like groups *V* _{ n }(*G*)

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## Abstract

We find some perhaps surprising isomorphism results for the groups {*V* _{ n }(*G*)}, where *V* _{ n }(*G*) is a supergroup of the Higman–Thompson group *V* _{ n } for *n* ∈ N and *G* ≤ *S* _{ n }, the symmetric group on *n* points. These groups, introduced by Farley and Hughes, are the groups generated by *V* _{ n } and the tree automorphisms [*α*]_{ g } defined as follows. For each *g* ∈ *G* and each node *α* in the infinite rooted *n*-ary tree, the automorphisms [*α*]_{ g } acts iteratively as *g* on the child leaves of *α* and every descendent of *α*. In particular, we show that *V* _{ n } ≅ *V* _{ n }(*G*) if and only if *G* is semiregular (acts freely on *n* points), as well as some additional sufficient conditions for isomorphisms between other members of this family of groups. Essential tools in the above work are a study of the dynamics of the action of elements of *V* _{ n }(*G*) on the Cantor space, Rubin’s Theorem, and transducers from Grigorchuk, Nekrashevych, and Suschanskiĭ’s rational group on the *n*-ary alphabet.

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