Asymptoticity of grafting and Teichmüller rays

Gupta, Subhojoy¹

¹ Division of Physics, Mathematics and Astronomy, California Institute of Technology, 1200 E. California Blvd., Mathematics 253-37, Pasadena, CA 91125, USA and Center for Quantum Geometry of Moduli Spaces, Ny Munkegade 118, Aarhus C 8000, Denmark

Geometry & topology, Tome 18 (2014) no. 4, pp. 2127-2188.

Résumé

We show that any grafting ray in Teichmüller space determined by an arational lamination or a multicurve is (strongly) asymptotic to a Teichmüller geodesic ray. As a consequence the projection of a generic grafting ray to the moduli space is dense. We also show that the set of points in Teichmüller space obtained by integer ($2\pi$–) graftings on any hyperbolic surface projects to a dense set in the moduli space. This implies that the conformal surfaces underlying complex projective structures with any fixed Fuchsian holonomy are dense in the moduli space.

Détail

DOI : 10.2140/gt.2014.18.2127

Classification : 30F60, 32G15, 57M50
Keywords: grafting rays, Teichmüller rays

Bibliographie

[1] L V Ahlfors, Lectures on quasiconformal mappings, University Lecture Series 38, Amer. Math. Soc. (2006)

[2] A Beurling, L Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956) 125

[3] C J Bishop, Bi-Lipschitz approximations of quasiconformal maps, Ann. Acad. Sci. Fenn. Math. 27 (2002) 97

[4] S A Bleiler, A J Casson, Automorphisms of surfaces after Nielsen and Thurston, London Math. Soc. Student Texts 9, Cambridge Univ. Press (1988)

[5] F Bonahon, Shearing hyperbolic surfaces, bending pleated surfaces and Thurston's symplectic form, Ann. Fac. Sci. Toulouse Math. 5 (1996) 233

[6] R D Canary, D B A Epstein, P Green, Notes on notes of Thurston, from: "Analytical and geometric aspects of hyperbolic space" (editor D B A Epstein), London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 3

[7] Y E Choi, D Dumas, K Rafi, Grafting rays fellow travel Teichmüller geodesics, Int. Math. Res. Not. 2012 (2012) 2445

[8] R Díaz, I Kim, Asymptotic behavior of grafting rays, Geom. Dedicata 158 (2012) 267

[9] D Dumas, The Schwarzian derivative and measured laminations on Riemann surfaces, Duke Math. J. 140 (2007) 203

[10] D Dumas, Complex projective structures, from: "Handbook of Teichmüller theory, Vol. II" (editor A Papadopoulos), IRMA Lect. Math. Theor. Phys. 13, Eur. Math. Soc. (2009) 455

[11] D Dumas, M Wolf, Projective structures, grafting and measured laminations, Geom. Topol. 12 (2008) 351

[12] D B A Epstein, A Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, from: "Fundamentals of hyperbolic geometry: Selected expositions" (editors R D Canary, D Epstein, A Marden), London Math. Soc. Lecture Note Ser. 328, Cambridge Univ. Press (2006) 117

[13] G Faltings, Real projective structures on Riemann surfaces, Compositio Math. 48 (1983) 223

[14] A Fathi, F Laudenbach, V Poénaru, Thurston's work on surfaces, Mathematical Notes 48, Princeton Univ. Press (2012)

[15] W M Goldman, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987) 297

[16] S Gupta, Asymptoticity of grafting and Teichmüller rays, II, to appear in Geom. Dedicata.

[17] S Gupta, On the asymptotic behavior of complex earthquakes and Teichmüller disks, to appear in AMS Contemp. Math. Proceedings

[18] J H Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics, Vol. 1, Matrix Editions (2006)

[19] J H Hubbard, H Masur, Quadratic differentials and foliations, Acta Math. 142 (1979) 221

[20] Y Imayoshi, M Taniguchi, An introduction to Teichmüller spaces, Springer, Tokyo (1992)

[21] N V Ivanov, Isometries of Teichmüller spaces from the point of view of Mostow rigidity, from: "Topology, ergodic theory, real algebraic geometry" (editors V Turaev, A Vershik), Amer. Math. Soc. Transl. Ser. 2 202, Amer. Math. Soc. (2001) 131

[22] Y Kamishima, S P Tan, Deformation spaces on geometric structures, from: "Aspects of low-dimensional manifolds" (editors Y Matsumoto, S Morita), Adv. Stud. Pure Math. 20, Kinokuniya, Tokyo (1992) 263

[23] M Kapovich, Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Birkhäuser (2009)

[24] R S Kulkarni, U Pinkall, A canonical metric for Möbius structures and its applications, Math. Z. 216 (1994) 89

[25] A Lenzhen, H Masur, Criteria for the divergence of pairs of Teichmüller geodesics, Geom. Dedicata 144 (2010) 191

[26] H Masur, Uniquely ergodic quadratic differentials, Comment. Math. Helv. 55 (1980) 255

[27] H Masur, Dense geodesics in moduli space, 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) 417

[28] H Masur, Interval exchange transformations and measured foliations, Ann. of Math. 115 (1982) 169

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

[30] C Pommerenke, Boundary behaviour of conformal maps, Grundl. Math. Wissen. 299, Springer, Berlin (1992)

[31] K P Scannell, M Wolf, The grafting map of Teichmüller space, J. Amer. Math. Soc. 15 (2002) 893

[32] K Strebel, Quadratic differentials, Ergeb. Math. Grenzgeb. 5, Springer, Berlin (1984)

[33] H Tanigawa, Grafting, harmonic maps and projective structures on surfaces, J. Differential Geom. 47 (1997) 399

[34] W P Thurston, Minimal stretch maps between hyperbolic surfaces,

[35] W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979)