## Groups and Dynamics Seminar

**Date: ** April 3, 2024

**Time: ** 3:00PM - 4:00PM

**Location: ** BLOC 123

**Speaker: **Yuri Bahturin, Memorial University of Newfoundland

**Title: ***Growth of subideals in free Lie algebras and subnormal subgroups in free groups*

**Abstract: **This talk is based on two papers of A. Olshanski (a paper about Lie algebras is joint with the speaker). The papers are devoted to the thorough study of the growth and cogrowth functions of subnormal (subideal) closures of finite sets of elements in the free groups (free Lie algebras). It is interesting that the study of the growth functions often allows one to give answers to the questions in purely abstract form. For example: Any nonzero finitely generated subalgebra of a nonabelian free Lie algebra is self-idealizing, that is, equal to its idealizer. Or: No proper finitely generated subalgebra K of a free Lie algebra L can contain a nonzero subideal H of L. Or else: Let H be a normal subgroup in a free group F with infinite factor group F/H and S a finite subset of H. Then the normal closure N of S in H contains no nontrivial normal subgroups of F.

In the group-theoretical paper, the author often refers to the work of R. Grigorchuk.