Hopficity of vertex-transitive generalized Baumslag–Solitar groups

N. V. Guskov^a, F. A. Dudkin<sup>b

a</sup> Novosibirsk State University, Novosibirsk, Russia
^b Sobolev Institute of Mathematics, Novosibirsk, Russia

Abstract: A generalized Baumslag–Solitar group (GBS group) is a finitely generated group $G$ acting on a tree in such a way that all vertex and edge stabilizers are infinite cyclic groups. If $G$ acts transitively on the set of vertices, we refer to it as a vertex-transitive GBS group (VTGBS group). A group is said to be Hopfian if every epimorphism from the group onto itself is an isomorphism. In this paper, we obtain several sufficient conditions for VTGBS groups to be non-Hopfian and describe epimorphisms of such groups.

Keywords: generalized Baumslag–Solitar group, Hopfian group, vertex-transitive action.

UDC: 512.54

MSC: 35R30

Received: 18.12.2024
Revised: 24.05.2025
Accepted: 26.05.2025

DOI: 10.33048/smzh.2025.66.407

 English version: Siberian Mathematical Journal, 2025, 66:4, 946–951