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2008 On the isomorphism problem for generalized Baumslag–Solitar groups
Matt Clay, Max Forester
Algebr. Geom. Topol. 8(4): 2289-2322 (2008). DOI: 10.2140/agt.2008.8.2289


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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Matt Clay. Max Forester. "On the isomorphism problem for generalized Baumslag–Solitar groups." Algebr. Geom. Topol. 8 (4) 2289 - 2322, 2008.


Received: 10 October 2007; Accepted: 6 November 2008; Published: 2008
First available in Project Euclid: 20 December 2017

zbMATH: 1191.20021
MathSciNet: MR2465742
Digital Object Identifier: 10.2140/agt.2008.8.2289

Primary: 20E08
Secondary: 20F10, 20F28

Rights: Copyright © 2008 Mathematical Sciences Publishers


Vol.8 • No. 4 • 2008
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