Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 8, Number 4 (2008), 2289-2322.
On the isomorphism problem for generalized Baumslag–Solitar groups
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.
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
Permanent link to this document
Digital Object Identifier
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]
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