Introduction to groups acting on regular rooted trees (1). ( notes)
In this seminar, we will explore groups acting on regular rooted trees. As we will see, these groups provide a rich source of examples with interesting properties in group theory and have been instrumental in solving important mathematical problems.
For instance, the first Grigorchuk group, introduced by Rostislav Grigorchuk in 1980, is one of the first known examples of an infinite, finitely generated periodic group, thereby providing a negative solution to the General Burnside Problem. It is also the first example of a group with intermediate growth, resolving Milnor’s Problem. Additionally, the Grigorchuk group is amenable but not elementary amenable, has a solvable word problem, is commensurable with its own direct product, is just-infinite, and possesses many other remarkable properties.
The aim of this seminar is to read and understand the papers "Dimension and Randomness in Groups Acting on Rooted Trees" by Miklós Abért and Bálint Virág and "Amenability via Random Walks" by Laurent Bartholdi and Bálint Virág. To achieve this, we will begin by introducing groups acting on regular rooted trees and then develop some of the core ideas that will later be required to understand the aforementioned papers.
See the programm for more details.
References for the reading course:
Research talks
Bass-Serre theory
Totally disconnected locally compact groups
Mixed Topics
Word Growth in Groups
Bass-Serre Theory and Profinite Analogues
p-Adic analytic pro-p groups
Invariant random subgroups
Probabilistic methods in group theory
Buildings