Mathematics, Department of


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A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Mark Brittenham. Lincoln, Nebraska: July, 2013

Copyright (c) 2013 Nathan Corwin


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 Z wr Z^2 as a subgroup of V, proving a conjecture of Bleak and Salazar-Diaz. 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 occur as subgroups of nV, a different generalization of V introduced by Matt Brin.

Adviser: Professor Mark Brittenham