Abstract: A finitely generated group G acting on a tree with all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag–Solitar group (GBS-group). By the Bass–Serre Theorem, G is representable as π 1 (A), the fundamental group of a graph of groups A whose vertex and edge groups are infinite cyclic. To each GBS-group G, we can associate a labeled graph A, a particular kind of a graph of groups. Such a labeled graph corresponds to an action of G on a tree and defines a presentation of G. Any GBS group can be obtained from infinite cyclic groups using free constructions: amalgamated free product and HNN-extension. Our goal is to tell about some recent results on GBS groups: description of the centralizer dimension, the problem of universal equivalence, K-residuality,connection with knot groups. Some open problems will be discussed at the end of the talk.

Cite: Dudkin F.A.
Generalized Baumslag–Solitar groups: properties, results, problems
Международная конференция «Мальцевские чтения» 20-24 Sep 2021