Some isomorphism results for Thompson-like groups V n (G)

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.