Embedding and Nonembedding Results for R. Thompson's Group V and Related Groups

Adviser: Professor Mark Brittenham 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 ≀ Z 2 as a subgroup of V , proving a conjecture of Bleak and Salazar-D` ıaz. 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. iv ACKNOWLEDGMENTS I am in debt to my entire committee. To Collin Bleak, thank you for the faith you showed in me almost from our first meeting, for sharing your love and excitement of math in general and for the dynamics of V in particular, and for encouragement throughout the process. I also want to thank you for finding the right balance of help and advice while still ensuring that the math was all mine. To Mark Brittenham, thank you for advising me despite my working in a subject a little ways from yours. I appreciate the time you gave me and the ability you have to always ask an insightful question. I also appreciate the help in editing this thesis. Any error that remains is surely from my neglect of your comments. Stephen Scott, who also gave advice and encouragement. I would also like to thank John Orr for the help at the beginning of my research. You told me a story when my committee was formed about how the research of your own that you are most proud of was just a partial result. This makes me feel better that my result is just a special case of the Theorem I want to prove. Outside my committee, the person who helped me the most is Susan Hermiller. You …