Algebra Seminar talk

Tim Herbstrith
Die erste Grigorchuk-Gruppe

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 20th 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.