Automata groups

Abstract

This dissertation is devoted to the groups generated by automata. The first part of the dissertation deals with L-presentations for such groups. We describe the sufficient condition for an essentially free automaton group to have an L-presentation. We also find the L-presentation for several other groups generated by three-state automata, and we describe the defining relations in the Grigorchuk groups G_w. In case when the sequence w is almost periodic these relations provide an L-presentation for the group G_w. We also describe defining relations in the series of groups which contain Grigorchuk-Erschler group and the group of iterated monodromies of the polynomial z^2 + i. The second part of the dissertation considers groups generated by 3-state automata over the alphabet of 2 letters and 2-state automata over the 3-letter alphabet. We continue the classification work started by the research group at Texas A&M University ([BGK+07a, BGK+07b]) and further reduce the number of pairwise nonisomorphic groups generated by 3-state automata over the 2-letter alphabet. We also study the groups generated by 2-state automata over the 3-letter alphabet and obtain a number of classification results for this class of group.