Self-similar Group Actions
by Tian Sang
Abstract: Self-similar group actions capture the idea of a group action that is repeated at all scales. This is achieved by considering actions on a rooted tree such that the action of the group repeats at all levels of the tree. Self-similar groups are defined based on the automorphisms of rooted binary trees, and one way to describe this is by an automaton. When we have a self-similar group acting on a binary tree, the associated Schreier graphs help us visualize the group structure. The Schreier graphs together give us a self-similarity graph, which gives interesting insights on geometric group theory, such as the hyperbolicity of the graph and contracting of the group.
This was the Vacation Scholarship project that I worked on with my supervisor Lawrence Reeves last summer. In this talk, I will explain how the self-similar group and self-similarity graph were defined and how to construct Schreier graphs from the given automaton for some concrete examples. I also hope there will be enough time for me to show some Schreier graphs I generated by Python and Sage, and how they provide insights on group structure.