[Different place and time!] Miloš Puđa: Self-similarity and amenability (via random walks)

This talk provides a leisurely introduction to the ideas, methods, and theory of self-similar groups. The class of self-similar groups consists of recursively defined automorphisms of rooted k-regular trees, but they can also be described using automata or wreath products. These groups also arise as the iterated monodromy groups of rational endomorphisms of the Riemann sphere, where the limiting Schreier graphs of the action on the tree’s level sets yield intriguing fractal sets. Consequently, self-similar groups contribute a wealth of exotic examples to the "algebraic zoo". We will also introduce and briefly discuss amenability, a group property that generalizes the concept of finiteness. We are particularly interested in the following characterization: a group is amenable if it admits a random walk with trivial behavior at infinity. Because random walks on self-similar groups are themselves self-similar, we can utilize this recursive structure to analyze their asymptotic behavior. We will show using these ideas that the iterated monodromy group IMG(z^2−1) (the so-called Basilica group) is amenable. Time permitting, we will discuss extending this result to all self-similar groups generated by automata of linear growth.