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On the isomorphism problem for generalized
Baumslag–Solitar groups
Matt Clay and Max Forester
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Algebraic & Geometric Topology 8
(2008) 2289–2322
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Abstract
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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.
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Keywords
generalized Baumslag–Solitar group,
G-tree, labeled graph, deformation, JSJ decomposition,
automorphism group
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Mathematical Subject Classification
Primary: 20E08
Secondary: 20F10, 20F28
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Publication
Received: 10 October 2007
Accepted: 6 November 2008
Published: 20 December 2008
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