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

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.