Self-similar Groups: Invariance and Structure

This lecture explores self-similar groups, focusing on their invariance under section maps and the structure they exhibit. It delves into the concept of self-similarity in group theory, discussing closures, topological structures, and metric properties. The lecture also covers the Grigorchuk group, examples of finitely constrained groups, and the characterization of sofic tree shift groups. Various theorems are presented, including conditions for a group to be a group tree-shift of finite type. The lecture concludes with insights into self-similar, self-replicating, and level-transitive groups, highlighting the closure properties in Aut(X*).