Groups and Semigroups Generated by Automata

In this dissertation we classify the metabelian groups arising from a restricted class of invertible synchronous automata over a binary alphabet. We give faithful, self-similar actions of Heisenberg groups and upper triangular matrix groups. We introduce a new class of semigroups given by a restricted class of asynchronous automata. We call these semigroups " expanding automaton semigroups ". We show that this class strictly contains the class of automaton semigroups, and we show that the class of asynchronous automaton semigroups strictly contains the class of expanding automaton semigroups. We demonstrate that undecidability arises in the actions of expanding automaton semigroups and semigroups arising from asynchronous automata on regular rooted trees. In particular, we show that one cannot decide whether or not an element of an asynchronous automaton semigroup has a fixed point. We show that expanding automaton semigroups are residually finite, while semigroups given by asynchronous automata need not be. We show that the class of expanding automaton semigroups is not closed under taking normal ideal extensions, but the class of semigroups given by asynchronous automata is closed under taking normal ideal extensions. We show that the class of expanding automaton semigroups is closed under taking direct products, provided that the direct product is finitely generated. We show that the class of automaton semigroups is not closed under passing to residually finite Rees quotients. We show that every partially commutative monoid is an automaton semigroup, and every partially commutative semigroup is an expanding automaton semigroup. iii ACKNOWLEDGMENTS I would like to express my gratitude to my advisors, Susan Hermiller and John Meakin, for all of their support and for the countless hours spent in weekly meetings and reading drafts of my papers. In addition, I'd like to thank them for all of their help with the lengthy job search; thanks for editing my job materials and helping me with various talks. Thanks especially to Susan for introducing me to Cayley graphs, and to John for introducing me to the Stallings foldings paper. The beauty of these concepts helped my third-year-graduate-student self decide to study groups and semigroups. I would also like to thank the other members of my advisory committee: thank Mark and Jamie for reading over my thesis, and I'd especially like to thank Jamie for his charming British accent. I'd like to thank all of my buddies in the UNL math department, even though spending time with many …