Connected components of the compactification of representation spaces of surface groups

Wolff, Maxime¹

¹ Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie - Paris 6, Case 247, 4 place Jussieu, Fr-75005 Paris, France

Geometry & topology, Tome 15 (2011) no. 3, pp. 1225-1295.

Résumé

The Thurston compactification of Teichmüller spaces has been generalised to many different representation spaces by Morgan, Shalen, Bestvina, Paulin, Parreau and others. In the simplest case of representations of fundamental groups of closed hyperbolic surfaces in $PSL\left(2,ℝ\right)$, we prove that this compactification behaves very badly: the nice behaviour of the Thurston compactification of the Teichmüller space contrasts with wild phenomena happening on the boundary of the other connected components of these representation spaces. We prove that it is more natural to consider a refinement of this compactification, which remembers the orientation of the hyperbolic plane. The ideal points of this compactification are oriented $ℝ$–trees, ie, $ℝ$–trees equipped with a planar structure.

Détail

DOI : 10.2140/gt.2011.15.1225

Keywords: $\mathbb{R}$–tree, Euler class, surface group, Teichmüller space, Thurston's compactification

Bibliographie

[1] R C Alperin, K N Moss, Complete trees for groups with a real-valued length function, J. London Math. Soc. $(2)$ 31 (1985) 55

[2] G Baumslag, On generalised free products, Math. Z. 78 (1962) 423

[3] R Benedetti, C Petronio, Lectures on hyperbolic geometry, Universitext, Springer (1992)

[4] M Bestvina, Degenerations of the hyperbolic space, Duke Math. J. 56 (1988) 143

[5] N Bourbaki, Éléments de mathématique. Topologie générale. Chapitres 1 à 4, Hermann (1971)

[6] D Calegari, Circular groups, planar groups, and the Euler class, from: "Proceedings of the Casson Fest" (editors C Gordon, Y Rieck), Geom. Topol. Monogr. 7, Geom. Topol. Publ. (2004) 431

[7] C Champetier, V Guirardel, Limit groups as limits of free groups, Israel J. Math. 146 (2005) 1

[8] M Coornaert, T Delzant, A Papadopoulos, Géométrie et théorie des groupes: les groupes hyperboliques de Gromov, Lecture Notes in Math. 1441, Springer (1990)

[9] M Culler, P B Shalen, Varieties of group representations and splittings of $3$–manifolds, Ann. of Math. $(2)$ 117 (1983) 109

[10] J Deblois, R P Kent Iv, Surface groups are frequently faithful, Duke Math. J. 131 (2006) 351

[11] A Fathi, F Laudenbach, V Poénaru, Editors, Travaux de Thurston sur les surfaces, Astérisque 66–67, Soc. Math. France (1979) 284

[12] J Fresnel, Méthodes modernes en géométrie, Act. Sci. et Ind. 1437, Hermann (1998)

[13] L Funar, M Wolff, Non-injective representations of a closed surface group into $\mathrm{PSL}(2,\mathbb R)$, Math. Proc. Cambridge Philos. Soc. 142 (2007) 289

[14] É Ghys, Classe d'Euler et minimal exceptionnel, Topology 26 (1987) 93

[15] É Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée, from: "The Lefschetz centennial conference, Part III (Mexico City, 1984)" (editor A Verjovsky), Contemp. Math. 58, Amer. Math. Soc. (1987) 81

[16] É Ghys, Groups acting on the circle, Enseign. Math. $(2)$ 47 (2001) 329

[17] É Ghys, P De La Harpe, editors, Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Math. 83, Birkhäuser (1990)

[18] W M Goldman, Discontinuous groups and the Euler class, PhD thesis, University of California Berkeley (1980)

[19] W M Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984) 200

[20] W M Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988) 557

[21] V Guirardel, Approximations of stable actions on $\mathbf{R}$–trees, Comment. Math. Helv. 73 (1998) 89

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

[23] N Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. $(3)$ 55 (1987) 59

[24] M Kapovich, B Leeb, On asymptotic cones and quasi-isometry classes of fundamental groups of $3$–manifolds, Geom. Funct. Anal. 5 (1995) 582

[25] S Katok, Fuchsian groups, Chicago Lectures in Math., Univ. of Chicago Press (1992)

[26] B Leeb, A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry, Bonner Math. Schriften 326, Univ. Bonn Math. Inst. (2000)

[27] H Masur, Ergodic actions of the mapping class group, Proc. Amer. Math. Soc. 94 (1985) 455

[28] S Matsumoto, Some remarks on foliated $S^1$ bundles, Invent. Math. 90 (1987) 343

[29] C T Mcmullen, Ribbon $\mathbb R$–trees and holomorphic dynamics on the unit disk, J. Topol. 2 (2009) 23

[30] J Milnor, On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958) 215

[31] J W Morgan, Group actions on trees and the compactification of the space of classes of $\mathrm{SO}(n,1)$–representations, Topology 25 (1986) 1

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

[33] A Parreau, Compactification d'espaces de représentations de groupes de type fini, preprint (2010)

[34] A Parreau, Espaces de représentations complètement réductibles, J. London Math. Soc. 83 (2011) 545

[35] F Paulin, Topologie de Gromov équivariante, structures hyperboliques et arbres réels, Invent. Math. 94 (1988) 53

[36] F Paulin, The Gromov topology on $\mathbf{R}$–trees, Topology Appl. 32 (1989) 197

[37] F Paulin, Sur la compactification de Thurston de l'espace de Teichmüller, from: "Géométries à courbure négative ou nulle, groupes discrets et rigidités" (editors L Bessières, A Parreau, B Rémy), Sémin. Congr. 18, Soc. Math. France (2009) 421

[38] V Poénaru, Groupes discrets, Lecture Notes in Math. 421, Springer (1974)

[39] Z Sela, Diophantine geometry over groups. I: Makanin–Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. 93 (2001) 31

[40] R K Skora, Splittings of surfaces, J. Amer. Math. Soc. 9 (1996) 605

[41] W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. $($N.S.$)$ 19 (1988) 417

[42] A Weil, Remarks on the cohomology of groups, Ann. of Math. $(2)$ 80 (1964) 149

[43] M Wolff, Sur les composantes exotiques des espaces d'actions de groupes de surfaces sur le plan hyperbolique, PhD thesis, Université de Grenoble I (2007)

[44] J W Wood, Bundles with totally disconnected structure group, Comment. Math. Helv. 46 (1971) 257