Algebraic & Geometric Topology

On the isomorphism problem for generalized Baumslag–Solitar groups

Matt Clay and Max Forester

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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.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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.

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  • H Bass, R Kulkarni, Uniform tree lattices, J. Amer. Math. Soc. 3 (1990) 843–902
  • M Clay, Deformation spaces of $G$–trees and automorphisms of Baumslag–Solitar groups, to appear in Groups Geom. Dyn.
  • M Clay, Deformation spaces of $G$–trees, PhD thesis, University of Utah (2006)
  • M Clay, M Forester, Whitehead moves for $G$–trees, to appear in Bull. London Math. Soc.
  • M Forester, Deformation and rigidity of simplicial group actions on trees, Geom. Topol. 6 (2002) 219–267
  • M Forester, On uniqueness of JSJ decompositions of finitely generated groups, Comment. Math. Helv. 78 (2003) 740–751
  • M Forester, Splittings of generalized Baumslag–Solitar groups, Geom. Dedicata 121 (2006) 43–59
  • N D Gilbert, J Howie, V Metaftsis, E Raptis, Tree actions of automorphism groups, J. Group Theory 3 (2000) 213–223
  • V Guirardel, G Levitt, Deformation spaces of trees, Groups Geom. Dyn. 1 (2007) 135–181
  • P H Kropholler, Baumslag–Solitar groups and some other groups of cohomological dimension two, Comment. Math. Helv. 65 (1990) 547–558
  • P H Kropholler, A note on centrality in $3$–manifold groups, Math. Proc. Cambridge Philos. Soc. 107 (1990) 261–266
  • G Levitt, Characterizing rigid simplicial actions on trees, from: “Geometric methods in group theory”, Contemp. Math. 372, Amer. Math. Soc. (2005) 27–33
  • G Levitt, On the automorphism group of generalized Baumslag–Solitar groups, Geom. Topol. 11 (2007) 473–515
  • G Levitt, Personal communication (2008)
  • M R Pettet, The automorphism group of a graph product of groups, Comm. Algebra 27 (1999) 4691–4708
  • K Whyte, The large scale geometry of the higher Baumslag–Solitar groups, Geom. Funct. Anal. 11 (2001) 1327–1343