Affine deformations of a three-holed sphere

Charette, Virginie¹ ; Drumm, Todd² ; Goldman, William³

¹ Département de mathématiques, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada
² Department of Mathematics, Howard University, Washington DC 20059, USA
³ Department of Mathematics, University of Maryland, College Park MD 20742, USA

Geometry & topology, Tome 14 (2010) no. 3, pp. 1355-1382.

Résumé

Associated to every complete affine 3–manifold $M$ with nonsolvable fundamental group is a noncompact hyperbolic surface $\Sigma$. We classify these complete affine structures when $\Sigma$ is homeomorphic to a three-holed sphere. In particular, for every such complete hyperbolic surface $\Sigma$, the deformation space identifies with two opposite octants in ${ℝ}^3$. Furthermore every $M$ admits a fundamental polyhedron bounded by crooked planes. Therefore $M$ is homeomorphic to an open solid handlebody of genus two. As an explicit application of this theory, we construct proper affine deformations of an arithmetic Fuchsian group inside $Sp\left(4,\mathbb{Z}\right)$.

Détail

DOI : 10.2140/gt.2010.14.1355

Keywords: hyperbolic surface, affine manifold, discrete group, fundamental polygon, fundamental polyhedron, proper action, Lorentz metric, Fricke space

Bibliographie

[1] H Abels, Properly discontinuous groups of affine transformations: a survey, Geom. Dedicata 87 (2001) 309

[2] I Agol, Tameness of hyperbolic 3–manifolds

[3] T Barbot, V Charette, T Drumm, W M Goldman, K Melnick, A primer on the $(2+1)$ Einstein universe, from: "Recent developments in pseudo–Riemannian geometry", ESI Lect. Math. Phys., Eur. Math. Soc., Zürich (2008) 179

[4] D Calegari, D Gabai, Shrinkwrapping and the taming of hyperbolic 3–manifolds, J. Amer. Math. Soc. 19 (2006) 385

[5] V Charette, Affine deformations of ultraideal triangle groups, Geom. Dedicata 97 (2003) 17

[6] V Charette, The affine deformation space of a rank two Schottky group: a picture gallery, Geom. Dedicata 122 (2006) 173

[7] V Charette, Non-proper affine actions of the holonomy group of a punctured torus, Forum Math. 18 (2006) 121

[8] V Charette, T Drumm, Strong marked isospectrality of affine Lorentzian groups, J. Differential Geom. 66 (2004) 437

[9] V Charette, T A Drumm, The Margulis invariant for parabolic transformations, Proc. Amer. Math. Soc. 133 (2005) 2439

[10] V Charette, T A Drumm, W Goldman, Stretching three-holed spheres and the Margulis invariant, from: "In the tradition of Ahlfors–Bers V", Contemp. Math. 510, Amer. Math. Soc. (2010) 61

[11] V Charette, T Drumm, W Goldman, M Morrill, Complete flat affine and Lorentzian manifolds, Geom. Dedicata 97 (2003) 187

[12] V Charette, W M Goldman, Affine Schottky groups and crooked tilings, from: "Crystallographic groups and their generalizations (Kortrijk, 1999)", Contemp. Math. 262, Amer. Math. Soc. (2000) 69

[13] T A Drumm, Examples of nonproper affine actions, Michigan Math. J. 39 (1992) 435

[14] T A Drumm, Fundamental polyhedra for Margulis space-times, Topology 31 (1992) 677

[15] T A Drumm, Linear holonomy of Margulis space-times, J. Differential Geom. 38 (1993) 679

[16] T A Drumm, W M Goldman, Complete flat Lorentz 3–manifolds with free fundamental group, Internat. J. Math. 1 (1990) 149

[17] T A Drumm, W M Goldman, The geometry of crooked planes, Topology 38 (1999) 323

[18] T A Drumm, W M Goldman, Isospectrality of flat Lorentz 3–manifolds, J. Differential Geom. 58 (2001) 457

[19] D Fried, W M Goldman, Three-dimensional affine crystallographic groups, Adv. in Math. 47 (1983) 1

[20] W M Goldman, The Margulis invariant of isometric actions on Minkowski $(2+1)$–space, from: "Rigidity in dynamics and geometry (Cambridge, 2000)", Springer (2002) 187

[21] W M Goldman, Trace coordinates on Fricke spaces of some simple hyperbolic surfaces, from: "Handbook of Teichmüller theory Vol II", IRMA Lect. Math. Theor. Phys. 13, Eur. Math. Soc., Zürich (2009) 611

[22] W M Goldman, F Labourie, G Margulis, Proper affine actions and geodesic flows of hyperbolic surfaces, Ann. Math. 170 (2009) 1051

[23] W M Goldman, G A Margulis, Flat Lorentz 3–manifolds and cocompact Fuchsian groups, from: "Crystallographic groups and their generalizations (Kortrijk, 1999)", Contemp. Math. 262, Amer. Math. Soc. (2000) 135

[24] W M Goldman, G A Margulis, Y Minsky, Complete flat Lorentz 3–manifolds and laminations of hyperbolic surfaces, in preparation

[25] C Jones, Pyramids of properness, PhD thesis, University of Maryland (2003)

[26] F Labourie, Fuchsian affine actions of surface groups, J. Differential Geom. 59 (2001) 15

[27] G A Margulis, Free completely discontinuous groups of affine transformations, Dokl. Akad. Nauk SSSR 272 (1983) 785

[28] G A Margulis, Complete affine locally flat manifolds with a free fundamental group, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 134 (1984) 190

[29] G Mess, Lorentz spacetimes of constant curvature, Geom. Dedicata 126 (2007) 3

[30] J Milnor, On fundamental groups of complete affinely flat manifolds, Advances in Math. 25 (1977) 178

[31] J G Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer (2006)