Strongly verbally closed groupsGeometric Group Theory Oberseminar

by Filipp Denissov (Universität Bonn)

Wednesday Jan 22, 2025, 11:00 AM → 12:00 PM Europe/Berlin

Endenicher Allee 60/0-003 - Seminarraum (Mathezentrum)

A subgroup H of a group G is called verbally closed, if any splitted equation over H, having solutions in G, has a solution in H. It is called algebraically closed, if any finite system of equations over H, having solutions in G, has a solution in H. It is difficult to verify these properties, but they often turn out to be equivalent. 

A group H is called strongly verbally closed, if it is an algebraically closed subgroup in any group containing H as a verbally closed subgroup. 

The class of strongly verbally closed groups is fairly wide. It includes all acylindrically hyperbolic groups, with no nontrivial finite normal subgroups (as showed by Bogopolski), all finite groups with non-abelian monoliths (as showed by Klyachko, Miroshnichenko, and Olshanskii), and many other groups. Among non-strongly-verbally-closed groups are non-abelian braid groups, and SL_2n (Z) (as showed by Denissov and Klyachko).

The purpose of the talk is to cover known results concerning strongly verbally closed groups.