# Algebra Seminar talk

2017-04-28

Tim **Herbstrith***Die erste Grigorchuk-Gruppe*

Abstract:

The subject matter of this talk is the first Grigorchuk group in context of
Burnside’s problems. Since this group, which was introduced by Rostislav
Grigorchuk (Ukr.: Григорчук) in 1980, is a subgroup of the automorphism group
of a full binary tree, in the first part of this talk I will introduce some
graph theoretical notions. The second part shows that the first Grigorchuk
group poses a counterexample to the unbounded Burnside problem, which was first
stated in a paper by William Burnside in 1902, and asks whether each finitely
generated periodic group is finite. Burnside’s question had significant impact
on the whole field of group theory throughout the 20^{th} century. In the last
part I will talk about the growth of the first Grigorchuk group with respect to
the word length. I will sketch a proof that the orders of the automorphisms in
this group cannot be bound uniformly. As a consequence, the first Grigorchuk
group does not pose a counterexample to the stronger bounded Burnside problem.

Depending on the preference of the audience this talk will be given in English or German.