Live Stream: https://youtube.com/live/jbUvo4xMU2g

Suppose we take $N$ steps on an orbital graph; each step corresponds to multiplying by a group generator, and we want to visit as many distinct sites as possible. If the graph is infinite, we can visit $N$ sites, so the maximal orbit growth is linear. Now we multiply the steps in inverse order. For the action of a group on itself, and for many other actions, the maximal inverted growth is linear.  We will discuss the somewhat counterintuitive phenomenon that the maximal inverted orbit growth can be sublinear. This leads to a new construction of groups of intermediate growth, in particular of many extensions of Grigorchuk groups. It allows characterizing countable groups which are subgroups of groups of intermediate growth, and improves our understanding of other questions about subgroups of such groups. Sublinear orbit growth is also behind Nekrashevych's construction of simple groups of intermediate growth, and the inverted growth of the first Grigorchuk group is an essential ingredient in recent results of Sauer and Schesler about finitely presented groups of non-uniform exponential growth.

Other topics are (self-similar and fractal) geometry of Schreier graphs, and various finiteness properties of groups. We will also continue our discussion of groups and semigroups generated by finite-state automata—a notion originating in computer science and the 1965 work of Krohn and Rhodes. This lecture will be self-contained, so prior attendance is not required.