JSJ decompositions of quadratic Baumslag–Solitar groups

Alonso, Juan¹

¹Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, NY 14853, USA

Algebraic and Geometric Topology, Tome 12 (2012) no. 4, pp. 2027-2047

Résumé

Generalized Baumslag–Solitar groups are defined as fundamental groups of graphs of groups with infinite cyclic vertex and edge groups. Forester [Geom. Topol. 6 (2002) 219-267] proved that in most cases the defining graphs are cyclic JSJ decompositions, in the sense of Rips and Sela. Here we extend Forester’s results to graphs of groups with vertex groups that can be either infinite cyclic or quadratically hanging surface groups.

Détail

DOI : 10.2140/agt.2012.12.2027

Classification : 20E06, 20E08, 20F65, 57M60
Keywords: JSJ decomposition, generalized Baumslag–Solitar groups, graphs of groups

Bibliographie

[1] M Bestvina, M Feighn, Bounding the complexity of simplicial group actions on trees, Invent. Math. 103 (1991) 449

[2] B H Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998) 145

[3] M J Dunwoody, M E Sageev, JSJ-splittings for finitely presented groups over slender groups, Invent. Math. 135 (1999) 25

[4] M J Dunwoody, E L Swenson, The algebraic torus theorem, Invent. Math. 140 (2000) 605

[5] M Forester, Deformation and rigidity of simplicial group actions on trees, Geom. Topol. 6 (2002) 219

[6] M Forester, On uniqueness of JSJ decompositions of finitely generated groups, Comment. Math. Helv. 78 (2003) 740

[7] K Fujiwara, P Papasoglu, JSJ-decompositions of finitely presented groups and complexes of groups, Geom. Funct. Anal. 16 (2006) 70

[8] V Guirardel, G Levitt, JSJ decompositions : definitions, existence and uniqueness I : The JSJ deformation space

[9] V Guirardel, G Levitt, JSJ decompositions : definitions, existence and uniqueness II : Compatibility and acylindricity

[10] V Guirardel, G Levitt, A general construction of JSJ decompositions, from: "Geometric group theory" (editors G N Arzhantseva, L Bartholdi, J Burillo, E Ventura), Trends Math., Birkhäuser (2007) 65

[11] W H Jaco, P B Shalen, Seifert fibered spaces in 3–manifolds, Mem. Amer. Math. Soc. 21 (1979)

[12] K Johannson, Homotopy equivalences of 3–manifolds with boundaries, 761, Springer (1979)

[13] P H Kropholler, An analogue of the torus decomposition theorem for certain Poincaré duality groups, Proc. London Math. Soc. 60 (1990) 503

[14] E Rips, Z Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. of Math. 146 (1997) 53

[15] P Scott, G A Swarup, Regular neighbourhoods and canonical decompositions for groups, 289, Soc. Math. France (2003)

[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

[17] J P Serre, Trees, Springer (1980)

[18] J Stallings, Group theory and 3–dimensional manifolds, 4, Yale Univ. Press (1971)

[19] H Wilton, One-ended subgroups of graphs of free groups with cyclic edge groups, Geom. Topol. 16 (2012) 665

[20] H Zieschang, E Vogt, H D Coldewey, Surfaces and planar discontinuous groups, 835, Springer (1980)