Algebraic & Geometric Topology

On the isomorphism problem for generalized Baumslag–Solitar groups

Matt Clay and Max Forester

Full-text: Open access

Abstract

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.

Article information

Source
Algebr. Geom. Topol., Volume 8, Number 4 (2008), 2289-2322.

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796935

Digital Object Identifier
doi:10.2140/agt.2008.8.2289

Mathematical Reviews number (MathSciNet)
MR2465742

Zentralblatt MATH identifier
1191.20021

Subjects
Primary: 20E08: Groups acting on trees [See also 20F65]
Secondary: 20F10: Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70] 20F28: Automorphism groups of groups [See also 20E36]

Keywords
generalized Baumslag–Solitar group G-tree labeled graph deformation JSJ decomposition automorphism group

Citation

Clay, Matt; Forester, Max. On the isomorphism problem for generalized Baumslag–Solitar groups. Algebr. Geom. Topol. 8 (2008), no. 4, 2289--2322. doi:10.2140/agt.2008.8.2289. https://projecteuclid.org/euclid.agt/1513796935


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