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

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 \(G\).


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
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[1] M Bestvina, M Feighn, Bounding the complexity of simplicial group actions on trees, Invent. Math. 103 (1991) 449 · Zbl 0724.20019
[2] B H Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998) 145 · Zbl 0911.57001
[3] M J Dunwoody, M E Sageev, JSJ-splittings for finitely presented groups over slender groups, Invent. Math. 135 (1999) 25 · Zbl 0939.20047
[4] M J Dunwoody, E L Swenson, The algebraic torus theorem, Invent. Math. 140 (2000) 605 · Zbl 1017.20034
[5] M Forester, Deformation and rigidity of simplicial group actions on trees, Geom. Topol. 6 (2002) 219 · Zbl 1118.20028
[6] M Forester, On uniqueness of JSJ decompositions of finitely generated groups, Comment. Math. Helv. 78 (2003) 740 · Zbl 1040.20032
[7] K Fujiwara, P Papasoglu, JSJ-decompositions of finitely presented groups and complexes of groups, Geom. Funct. Anal. 16 (2006) 70 · Zbl 1097.20037
[8] V Guirardel, G Levitt, JSJ decompositions: definitions, existence and uniqueness I: The JSJ deformation space · Zbl 1162.20017
[9] V Guirardel, G Levitt, JSJ decompositions: definitions, existence and uniqueness II: Compatibility and acylindricity · Zbl 1162.20017
[10] V Guirardel, G Levitt, A general construction of JSJ decompositions (editors G N Arzhantseva, L Bartholdi, J Burillo, E Ventura), Trends Math., Birkhäuser (2007) 65 · Zbl 1162.20017
[11] W H Jaco, P B Shalen, Seifert fibered spaces in \(3\)-manifolds, Mem. Amer. Math. Soc. 21 (1979) · Zbl 0415.57005
[12] K Johannson, Homotopy equivalences of \(3\)-manifolds with boundaries, Lecture Notes in Mathematics 761, Springer (1979) · Zbl 0412.57007
[13] P H Kropholler, An analogue of the torus decomposition theorem for certain Poincaré duality groups, Proc. London Math. Soc. 60 (1990) 503 · Zbl 0704.20023
[14] E Rips, Z Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. of Math. 146 (1997) 53 · Zbl 0910.57002
[15] P Scott, G A Swarup, Regular neighbourhoods and canonical decompositions for groups, Astérisque 289, Soc. Math. France (2003) · Zbl 1036.20028
[16] Z Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank \(1\) Lie groups. II, Geom. Funct. Anal. 7 (1997) 561 · Zbl 0884.20025
[17] J P Serre, Trees, Springer (1980) · Zbl 0548.20018
[18] J Stallings, Group theory and \(3\)-dimensional manifolds, Yale Math. Monogr. 4, Yale Univ. Press (1971) · Zbl 0241.57001
[19] H Wilton, One-ended subgroups of graphs of free groups with cyclic edge groups, Geom. Topol. 16 (2012) 665 · Zbl 1248.20047
[20] H Zieschang, E Vogt, H D Coldewey, Surfaces and planar discontinuous groups, Lecture Notes in Mathematics 835, Springer (1980) · Zbl 0438.57001
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