Dynamics of the mapping class group action on the variety of PSL2ℂ characters

Souto, Juan¹ ; Storm, Peter²

¹ Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago IL 60637-1514, USA
² Department of Mathematics, Stanford University, 450 Serra Mall, Stanford CA 94305-2125, USA

Geometry & topology, Tome 10 (2006) no. 2, pp. 715-736.

Résumé

We study the action of the mapping class group $Mod\left(S\right)$ on the boundary $\partial \mathcal{Q}$ of quasifuchsian space $\mathcal{Q}$. Among other results, $Mod\left(S\right)$ is shown to be topologically transitive on the subset $\mathcal{C}\subset \partial \mathcal{Q}$ of manifolds without a conformally compact end. We also prove that any open subset of the character variety $\mathcal{X}\left({\pi }_1\left(S\right),{SL}_2ℂ\right)$ intersecting $\partial \mathcal{Q}$ does not admit a nonconstant $Mod\left(S\right)$–invariant meromorphic function. This is related to a question of Goldman.

Détail

DOI : 10.2140/gt.2006.10.715

Keywords: hyperbolic geometry, mapping class group

Bibliographie

[1] J W Anderson, A brief survey of the deformation theory of Kleinian groups, from: "The Epstein birthday schrift (Warwick 1997)" (editors I Rivin, C Rourke, C Series), Geom. Topol. Monogr. 1 (1998) 23

[2] J W Anderson, R D Canary, Algebraic limits of Kleinian groups which rearrange the pages of a book, Invent. Math. 126 (1996) 205

[3] F Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. $(2)$ 124 (1986) 71

[4] F Bonahon, J P Otal, Variétés hyperboliques à géodésiques arbitrairement courtes, Bull. London Math. Soc. 20 (1988) 255

[5] J Brock, K Bromberg, R Evans, J Souto, Tameness on the boundary and Ahlfors' measure conjecture, Publ. Math. Inst. Hautes Études Sci. (2003) 145

[6] R D Canary, M Culler, S Hersonsky, P B Shalen, Approximation by maximal cusps in boundaries of deformation spaces of Kleinian groups, J. Differential Geom. 64 (2003) 57

[7] R D Canary, S Hersonsky, Ubiquity of geometric finiteness in boundaries of deformation spaces of hyperbolic 3–manifolds, Amer. J. Math. 126 (2004) 1193

[8] T Comar, Hyperbolic Dehn surgery and convergence of Kleinian groups, PhD thesis, University of Michigan (1996)

[9] M Culler, Lifting representations to covering groups, Adv. in Math. 59 (1986) 64

[10] R Evans, Weakly type-preserving sequences and strong convergence, Geom. Dedicata 108 (2004) 71

[11] W M Goldman, Ergodic theory on moduli spaces, Ann. of Math. $(2)$ 146 (1997) 475

[12] W M Goldman, The complex-symplectic geometry of $\mathrm{SL}(2,\mathbb C)$-characters over surfaces, from: "Algebraic groups and arithmetic", Tata Inst. Fund. Res. (2004) 375

[13] M Heusener, J Porti, The variety of characters in $\mathrm{PSL}_2(\mathbb C)$, Bol. Soc. Mat. Mexicana $(3)$ 10 (2004) 221

[14] M Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics 183, Birkhäuser (2001)

[15] S P Kerckhoff, W P Thurston, Noncontinuity of the action of the modular group at Bers' boundary of Teichmüller space, Invent. Math. 100 (1990) 25

[16] C Mcmullen, Cusps are dense, Ann. of Math. $(2)$ 133 (1991) 217

[17] C T Mcmullen, Renormalization and 3–manifolds which fiber over the circle, Annals of Mathematics Studies 142, Princeton University Press (1996)

[18] C T Mcmullen, Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998) 283

[19] J P Otal, Thurston's hyperbolization of Haken manifolds, from: "Surveys in differential geometry, Vol. III (Cambridge, MA, 1996)", Int. Press, Boston (1998) 77

[20] D Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984) 259

[21] D Sullivan, Quasiconformal homeomorphisms and dynamics II: Structural stability implies hyperbolicity for Kleinian groups, Acta Math. 155 (1985) 243

[22] W Thurston, The geometry and topology of 3–manifolds, Princeton University lecture notes (1976–1979)

[23] W Thurston, Hyperbolic structures on 3–manifolds II: Surface groups and 3–manifolds which fiber over the circle (1986)

[24] P Tukia, The Hausdorff dimension of the limit set of a geometrically finite Kleinian group, Acta Math. 152 (1984) 127