Quotients and subgroups of Baumslag-Solitar groups

TL;DR

This paper classifies all generalized Baumslag-Solitar groups that are quotients of a given Baumslag-Solitar group and explores embeddings among these groups, providing a comprehensive understanding of their subgroup and quotient structures.

Contribution

It offers a complete characterization of quotients and embeddings of Baumslag-Solitar groups and identifies conditions for embedding specific groups into these structures.

Findings

Identifies all generalized Baumslag-Solitar groups that are quotients of BS(m,n)

Characterizes when BS(r,s) can be embedded into BS(m,n)

Determines which finitely generated groups embed into some BS(n,n)

Abstract

We determine all generalized Baumslag-Solitar groups (finitely generated groups acting on a tree with all stabilizers infinite cyclic) which are quotients of a given Baumslag-Solitar group BS(m,n), and (when BS(m,n) is not Hopfian) which of them also admit BS(m,n) as a quotient. We determine for which values of r,s one may embed BS(r,s) into a given BS(m,n), and we characterize finitely generated groups which embed into some BS(n,n).