Abstract:

A finitely generated group $G_n$ that acts on a tree $T$ such that all edge stabilizers are infinite cyclic groups and all vertex stabilizers are free Abelian groups of rank $n$ will be called a generalized Baumslag--Solitar group of type (n,1) ($GBS(n,1)$ group). In this paper we find a criterion for such groups to have a non-trivial center and prove that if $n\geqslant 3$ and two such groups with non-trivial center are isomorphic, then the corresponding $GBS(1,1)$ groups must also be isomorphic.