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On the isomorphism problem for generalized Baumslag–Solitar groups

Matt Clay and Max Forester

Algebraic & Geometric Topology 8 (2008) 2289–2322

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.

generalized Baumslag–Solitar group, G-tree, labeled graph, deformation, JSJ decomposition, automorphism group
Mathematical Subject Classification 2000
Primary: 20E08
Secondary: 20F10, 20F28
Received: 10 October 2007
Accepted: 6 November 2008
Published: 20 December 2008
Matt Clay
Mathematics Department
University of Oklahoma
Norman, OK 73019
Max Forester
Mathematics Department
University of Oklahoma
Norman, OK 73019