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Some isomorphism results for Thompson-like groups Vn(G) 

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Author(s)

Collin Bleak, Casey Donoven, Julius Jonušas

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Abstract

We find some perhaps surprising isomorphism results for the groups {Vn(G)}, where Vn(G) is a supergroup of the Higman–Thompson group Vn for n ∈ N and G ≤ Sn, the symmetric group on n points. These groups, introduced by Farley and Hughes, are the groups generated by Vn 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 Vn ≅ Vn(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 Vn(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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Original languageEnglish
Pages (from-to)1-19
JournalIsrael Journal of Mathematics
Volume222
Issue number1
DOIs
StatePublished - 8 Nov 2017

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