Trees of cylinders and canonical splittings

Guirardel, Vincent¹ ; Levitt, Gilbert²

¹ Institut de Mathématiques de Toulouse, Université de Toulouse CNRS (UMR 5219), 118 route de Narbonne, F-31062 Toulouse cedex 9, France, Institut de Recherche Mathématiques de Rennes, Université de Rennes 1 CNRS (UMR 6625), 263 avenue du General Leclerc, CS 74205, 35042 Rennes Cedex, France
² Laboratoire de Mathématiques Nicolas Oresme, Université de Caen CNRS (UMR 6139), BP 5186, F-14032 Caen Cedex, France

Geometry & topology, Tome 15 (2011) no. 2, pp. 977-1012.

Résumé

Let $T$ be a tree with an action of a finitely generated group $G$. Given a suitable equivalence relation on the set of edge stabilizers of $T$ (such as commensurability, coelementarity in a relatively hyperbolic group, or commutation in a commutative transitive group), we define a tree of cylinders $T_c$. This tree only depends on the deformation space of $T$; in particular, it is invariant under automorphisms of $G$ if $T$ is a JSJ splitting. We thus obtain $Out\left(G\right)$–invariant cyclic or abelian JSJ splittings. Furthermore, $T_c$ has very strong compatibility properties (two trees are compatible if they have a common refinement).

Détail

DOI : 10.2140/gt.2011.15.977

Keywords: JSJ decomposition, canonical decomposition, amalgamated free product

Bibliographie

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

[2] I Bumagin, O Kharlampovich, A Miasnikov, The isomorphism problem for finitely generated fully residually free groups, J. Pure Appl. Algebra 208 (2007) 961

[3] M Carette, The automorphism group of accessible groups, to appear in J. London Math. Soc.

[4] C Champetier, L'espace des groupes de type fini, Topology 39 (2000) 657

[5] M Clay, Contractibility of deformation spaces of $G$–trees, Algebr. Geom. Topol. 5 (2005) 1481

[6] M Clay, M Forester, Whitehead moves for $G$–trees, Bull. Lond. Math. Soc. 41 (2009) 205

[7] M Culler, K Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986) 91

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

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

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

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

[12] V Guirardel, Reading small actions of a one-ended hyperbolic group on $\bold R$–trees from its JSJ splitting, Amer. J. Math. 122 (2000) 667

[13] V Guirardel, Limit groups and groups acting freely on $\mathbb R^n$–trees, Geom. Topol. 8 (2004) 1427

[14] V Guirardel, G Levitt, Deformation spaces of trees, Groups Geom. Dyn. 1 (2007) 135

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

[16] V Guirardel, G Levitt, Scott and Swarup's regular neighborhood as a tree of cylinders, Pacific J. Math. 245 (2010) 79

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

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

[19] V Guirardel, G Levitt, Splittings and automorphisms of relatively hyperbolic groups, in preparation

[20] G Levitt, Automorphisms of hyperbolic groups and graphs of groups, Geom. Dedicata 114 (2005) 49

[21] J W Morgan, P B Shalen, Valuations, trees, and degenerations of hyperbolic structures. I, Ann. of Math. $(2)$ 120 (1984) 401

[22] D V Osin, Elementary subgroups of relatively hyperbolic groups and bounded generation, Internat. J. Algebra Comput. 16 (2006) 99

[23] P Papasoglu, E Swenson, Boundaries and JSJ decompositions of $\mathrm{CAT}(0)$–groups, Geom. Funct. Anal. 19 (2009) 559

[24] F Paulin, Sur la théorie élémentaire des groupes libres (d'après Sela), Astérisque 294, Soc. Math. France (2004) 363

[25] E Rips, Z Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. of Math. $(2)$ 146 (1997) 53

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

[27] Z Sela, Acylindrical accessibility for groups, Invent. Math. 129 (1997) 527

[28] Z Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank $1$ Lie groups. II, Geom. Funct. Anal. 7 (1997) 561

[29] Z Sela, Diophantine geometry over groups. VII. The elementary theory of a hyperbolic group, Proc. Lond. Math. Soc. $(3)$ 99 (2009) 217