Translation surfaces and the curve graph in genus two

Nguyen, Duc-Manh¹

¹Institut de Mathématiques de Bordeaux, Université de Bordeaux, CNRS UMR 5251, 351, Cours de la Libération, F-33405 Talence Cedex, France

Algebraic and Geometric Topology, Tome 17 (2017) no. 4, pp. 2177-2237

Résumé

Let S be a (topological) compact closed surface of genus two. We associate to each translation surface (X,ω) ∈ Ωℳ2 = ℋ(2) ⊔ℋ(1,1) a subgraph Ĉcyl of the curve graph of S. The vertices of this subgraph are free homotopy classes of curves which can be represented either by a simple closed geodesic or by a concatenation of two parallel saddle connections (satisfying some additional properties) on X. The subgraph Ĉcyl is by definition GL+(2, ℝ)–invariant. Hence it may be seen as the image of the corresponding Teichmüller disk in the curve graph. We will show that Ĉ cyl is always connected and has infinite diameter. The group Aff+(X,ω) of affine automorphisms of (X,ω) preserves naturally Ĉ cyl, we show that Aff+(X,ω) is precisely the stabilizer of Ĉ cyl in Mod(S). We also prove that Ĉ cyl is Gromov-hyperbolic if (X,ω) is completely periodic in the sense of Calta.

It turns out that the quotient of Ĉ cyl by Aff+(X,ω) is closely related to McMullen’s prototypes in the case that (X,ω) is a Veech surface in ℋ(2). We finally show that this quotient graph has finitely many vertices if and only if (X,ω) is a Veech surface for (X,ω) in both strata ℋ(2) and ℋ(1,1).

Détail

DOI : 10.2140/agt.2017.17.2177

Classification : 51H20, 54H15
Keywords: translation surface, curve complex, Gromov hyperbolicity

Bibliographie

[1] T Aougab, Uniform hyperbolicity of the graphs of curves, Geom. Topol. 17 (2013) 2855

[2] M Bestvina, K Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002) 69 | DOI

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

[4] F Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988) 139 | DOI

[5] B H Bowditch, Hyperbolic 3–manifolds and the geometry of the curve complex, from: "European Congress of Mathematics" (editor A Laptev), Eur. Math. Soc. (2005) 103

[6] B H Bowditch, Intersection numbers and the hyperbolicity of the curve complex, J. Reine Angew. Math. 598 (2006) 105 | DOI

[7] B H Bowditch, Uniform hyperbolicity of the curve graphs, Pacific J. Math. 269 (2014) 269 | DOI

[8] J F Brock, R D Canary, Y N Minsky, The classification of Kleinian surface groups, II : The ending lamination conjecture, Ann. of Math. 176 (2012) 1 | DOI

[9] P Buser, Geometry and spectra of compact Riemann surfaces, 106, Birkhäuser (1992)

[10] K Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc. 17 (2004) 871 | DOI

[11] K Calta, J Smillie, Algebraically periodic translation surfaces, J. Mod. Dyn. 2 (2008) 209 | DOI

[12] M Clay, K Rafi, S Schleimer, Uniform hyperbolicity of the curve graph via surgery sequences, Algebr. Geom. Topol. 14 (2014) 3325 | DOI

[13] M Duchin, C J Leininger, K Rafi, Length spectra and degeneration of flat metrics, Invent. Math. 182 (2010) 231 | DOI

[14] B Farb, D Margalit, A primer on mapping class groups, 49, Princeton Univ. Press (2012)

[15] R H Gilman, On the definition of word hyperbolic groups, Math. Z. 242 (2002) 529 | DOI

[16] U Hamenstädt, Train tracks and the Gromov boundary of the complex of curves, from: "Spaces of Kleinian groups" (editors Y N Minsky, M Sakuma, C Series), London Math. Soc. Lecture Note Ser. 329, Cambridge Univ. Press (2006) 187

[17] U Hamenstädt, Geometry of the complex of curves and of Teichmüller space, from: "Handbook of Teichmüller theory, I" (editor A Papadopoulos), IRMA Lect. Math. Theor. Phys. 11, Eur. Math. Soc. (2007) 447 | DOI

[18] U Hamenstädt, Stability of quasi-geodesics in Teichmüller space, Geom. Dedicata 146 (2010) 101 | DOI

[19] J L Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986) 157 | DOI

[20] W J Harvey, Boundary structure of the modular group, from: "Riemann surfaces and related topics: proceedings of the 1978 Stony Brook conference" (editors I Kra, B Maskit), Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 245

[21] S Hensel, P Przytycki, R C H Webb, 1–slim triangles and uniform hyperbolicity for arc graphs and curve graphs, J. Eur. Math. Soc. 17 (2015) 755 | DOI

[22] P Hubert, E Lanneau, Veech groups without parabolic elements, Duke Math. J. 133 (2006) 335 | DOI

[23] P Hubert, T A Schmidt, Infinitely generated Veech groups, Duke Math. J. 123 (2004) 49 | DOI

[24] N V Ivanov, Automorphism of complexes of curves and of Teichmüller spaces, Internat. Math. Res. Not. 1997 (1997) 651 | DOI

[25] R Kenyon, J Smillie, Billiards on rational-angled triangles, Comment. Math. Helv. 75 (2000) 65 | DOI

[26] S Kerckhoff, H Masur, J Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. 124 (1986) 293 | DOI

[27] E Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space, preprint (1999)

[28] M Kontsevich, A Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003) 631 | DOI

[29] E Lanneau, D M Nguyen, Connected components of Prym eigenform loci in genus three, Math. Ann. (2017) | DOI

[30] R Lehnert, On the critical exponent of infinitely generated Veech groups, Math. Ann. (2016) | DOI

[31] F Luo, Automorphisms of the complex of curves, Topology 39 (2000) 283 | DOI

[32] H Masur, Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J. 53 (1986) 307 | DOI

[33] H A Masur, Y N Minsky, Geometry of the complex of curves, I : Hyperbolicity, Invent. Math. 138 (1999) 103 | DOI

[34] H Masur, S Schleimer, The geometry of the disk complex, J. Amer. Math. Soc. 26 (2013) 1 | DOI

[35] H Masur, S Tabachnikov, Rational billiards and flat structures, from: "Handbook of dynamical systems, 1A" (editors B Hasselblatt, A Katok), North-Holland (2002) 1015 | DOI

[36] C T Mcmullen, Teichmüller geodesics of infinite complexity, Acta Math. 191 (2003) 191 | DOI

[37] C T Mcmullen, Teichmüller curves in genus two : discriminant and spin, Math. Ann. 333 (2005) 87 | DOI

[38] C T Mcmullen, Dynamics of SL2(R) over moduli space in genus two, Ann. of Math. 165 (2007) 397 | DOI

[39] Y N Minsky, Teichmüller geodesics and ends of hyperbolic 3–manifolds, Topology 32 (1993) 625 | DOI

[40] M Möller, Affine groups of flat surfaces, from: "Handbook of Teichmüller theory, II" (editor A Papadopoulos), IRMA Lect. Math. Theor. Phys. 13, Eur. Math. Soc. (2009) 369 | DOI

[41] D M Nguyen, Parallelogram decompositions and generic surfaces in Hhyp(4), Geom. Topol. 15 (2011) 1707 | DOI

[42] D M Nguyen, On the topology of H(2), Groups Geom. Dyn. 8 (2014) 513 | DOI

[43] K Rafi, Hyperbolicity in Teichmüller space, Geom. Topol. 18 (2014) 3025 | DOI

[44] M Rees, An alternative approach to the ergodic theory of measured foliations on surfaces, Ergodic Theory Dynamical Systems 1 (1981) 461

[45] J Smillie, The dynamics of billiard flows in rational polygons, from: "Dynamical systems, ergodic theory and applications" (editor Y G Sinai), Encyclopaedia of Mathematical Sciences 100, Springer (2000) 360

[46] J Smillie, B Weiss, Minimal sets for flows on moduli space, Israel J. Math. 142 (2004) 249 | DOI

[47] J Smillie, B Weiss, Characterizations of lattice surfaces, Invent. Math. 180 (2010) 535 | DOI

[48] M Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793 | DOI

[49] W A Veech, Geometric realizations of hyperelliptic curves, from: "Algorithms, fractals, and dynamics" (editor Y Takahashi), Plenum (1995) 217

[50] Y Vorobets, Periodic geodesics on generic translation surfaces, from: "Algebraic and topological dynamics" (editors S Kolyada, Y Manin, T Ward), Contemp. Math. 385, Amer. Math. Soc. (2005) 205 | DOI

[51] A Wright, Cylinder deformations in orbit closures of translation surfaces, Geom. Topol. 19 (2015) 413 | DOI

[52] A Wright, Translation surfaces and their orbit closures: an introduction for a broad audience, EMS Surv. Math. Sci. 2 (2015) 63 | DOI

[53] A Zorich, Flat surfaces, from: "Frontiers in number theory, physics, and geometry, I" (editors P Cartier, B Julia, P Moussa, P Vanhove), Springer (2006) 437 | DOI