Algebraic and Geometric Topology

Generalized Baumslag-Solitar groups (GBS groups) are groups that act
on trees with infinite cyclic edge and vertex stabilizers. Such an
action is described by a *labeled graph* (essentially, the
quotient graph of groups). This paper addresses the problem of
determining whether two given labeled graphs define isomorphic groups;
this is the *isomorphism problem* for GBS groups. There are two
main results and some applications. First, we find necessary and
sufficient conditions for a GBS group to be represented by only
finitely many reduced labeled graphs. These conditions can be checked
effectively from any labeled graph. Then we show that the isomorphism
problem is solvable for GBS groups whose labeled graphs have first Betti
number at most one.