Universal equivalence of generalized Baumslag–Solitar groups

A finitely generated group acting on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag–Solitar group (a $GBS$-group). Every $GBS$-group is the fundamental group $\pi_1(\mathbb{A})$ of a suitable labeled graph $\mathbb{A}$. We prove that if $\mathbb{A}$ and $\mathbb{B}$ are labeled trees, then the groups $\pi_1(\mathbb{A})$ and $\pi_1(\mathbb{B})$ are universally equivalent iff $\pi_1(\mathbb{A})$ and $\pi_1(\mathbb{B})$ are embeddable into each other. An algorithm for verifying universal equivalence is pointed out. Moreover, we specify simple conditions for checking this criterion in the case where the centralizer dimension is equal to $3$.