JSJ decompositions of quadratic Baumslag-Solitar groups. (English) Zbl 1267.20042

Alonso, Juan

Algebr. Geom. Topol. 12, No. 4, 2027-2047 (2012).

A generalized Baumslag-Solitar (GBS) graph is a graph of groups whose edge and vertex groups are infinite cyclic. A quadratic Baumslag-Solitar (QBS) graph is a graph of groups whose edge groups are infinite cyclic and whose vertex groups are either infinite cyclic or are quadratically hanging. (The definition of quadratically hanging vertex groups used here differs slightly from the one originally used by Rips and Sela.)
Forester showed that (under mild hypotheses) if \(G\) is the fundamental group of a GBS graph \(\Gamma\) then \(\Gamma\) is a JSJ decomposition of \(G\). Here, this result is extended by showing that (under certain natural hypotheses) if \(G\) is the fundamental group of a QBS graph \(\Gamma\) then \(\Gamma\) is a Rips-Sela JSJ decomposition for \(G\).

Reviewer: Gerald Williams (Colchester)

MSC:

20E06	Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20E08	Groups acting on trees
20F65	Geometric group theory
57M60	Group actions on manifolds and cell complexes in low dimensions
20F05	Generators, relations, and presentations of groups

Keywords:

JSJ decompositions; generalized Baumslag-Solitar groups; graphs of groups

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