Embedding of baumslag-solitar groups into the generalized Baumslag-Solitar groups

Abstract

A finitely generated group G that acts on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag-Solitar group or GBS-group. Let p and q be coprime integers other than 0, 1, and −1. We prove that the Baumslag-Solitar group BS(p, q) embeds into G if and only if the equation x−1ypx = yq is solvable in G for y ≠ 1; i.e., \(\tfrac{p} {q} \) ∈ Δ(G), where Δ is the modular homomorphism.