Embedding and nonembedding results for R. Thompson's group V and related groups

Abstract

We study Richard Thompson's group V, and some generalizations of this group. V was one of the first two examples of a finitely presented, infinite, simple group. Since being discovered in 1965, V has appeared in a wide range of mathematical subjects. Despite many years of study, much of the structure of V remains unclear. Part of the difficulty is that the standard presentation for V is complicated, hence most algebraic techniques have yet to prove fruitful.

This thesis obtains some further understanding of the structure of V by showing the nonexistence of the wreath product [special characters omitted] as a subgroup of V, proving a conjecture of Bleak and Salazar-Dìaz. This result is achieved primarily by studying the topological dynamics occurring when V acts on the Cantor Set. We then show the same result for one particular generalization of V, the Higman-Thompson Groups G_n,r. In addition we show that some other wreath products do occcur as subgroups of nV, a different generalization of V introduced by Matt Brin.