2π–grafting and complex projective structures, I

Baba, Shinpei¹

¹ Ruprecht-Karls-Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 368, Heidelberg, D-69120, Germany

Geometry & topology, Tome 19 (2015) no. 6, pp. 3233-3287.

Résumé

Let $S$ be a closed oriented surface of genus at least two. Gallo, Kapovich and Marden asked whether $2\pi$–grafting produces all projective structures on $S$ with arbitrarily fixed holonomy (the Grafting conjecture). In this paper, we show that the conjecture holds true “locally” in the space $\mathcal{G}\mathcal{ℒ}$ of geodesic laminations on $S$ via a natural projection of projective structures on $S$ into $\mathcal{G}\mathcal{ℒ}$ in Thurston coordinates. In a sequel paper, using this local solution, we prove the conjecture for generic holonomy.

Détail

DOI : 10.2140/gt.2015.19.3233

Classification : 57M50, 30F40, 20H10
Keywords: surface, complex projective structure, holonomy, grafting

Bibliographie

[1] S Baba, A Schottky decomposition theorem for complex projective structures, Geom. Topol. 14 (2010) 117

[2] S Baba, Complex projective structures with Schottky holonomy, Geom. Funct. Anal. 22 (2012) 267

[3] F Bonahon, Variations of the boundary geometry of $3$–dimensional hyperbolic convex cores, J. Differential Geom. 50 (1998) 1

[4] J F Brock, Continuity of Thurston's length function, Geom. Funct. Anal. 10 (2000) 741

[5] G Calsamiglia, B Deroin, S Francaviglia, The oriented graph of multi-graftings in the Fuchsian case, Publ. Mat. 58 (2014) 31

[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), LMS Lecture Note Ser. 111, Cambridge Univ. Press (1987) 3

[7] R D Canary, D B A Epstein, P L Green, Notes on notes of Thurston, from: "Fundamentals of hyperbolic geometry: Selected expositions" (editors R D Canary, D Epstein, A Marden), LMS Lecture Note Ser. 328, CUP (2006) 1

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

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

[10] C J Earle, On variation of projective structures, from: "Riemann surfaces and related topics" (editors I Kra, B Maskit), Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 87

[11] D B A Epstein, A Marden, V Markovic, Quasiconformal homeomorphisms and the convex hull boundary, Ann. of Math. 159 (2004) 305

[12] D Gallo, M Kapovich, A Marden, The monodromy groups of Schwarzian equations on closed Riemann surfaces, Ann. of Math. 151 (2000) 625

[13] W M Goldman, Discontinuous groups and the Euler class, PhD thesis, Univ. California, Berkeley (1980)

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

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

[16] D A Hejhal, Monodromy groups and linearly polymorphic functions, Acta Math. 135 (1975) 1

[17] J H Hubbard, The monodromy of projective structures, from: "Riemann surfaces and related topics" (editors I Kra, B Maskit), Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 257

[18] K Ito, Exotic projective structures and quasi-Fuchsian space, II, Duke Math. J. 140 (2007) 85

[19] K Ito, On continuous extensions of grafting maps, Trans. Amer. Math. Soc. 360 (2008) 3731

[20] 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 (1992) 263

[21] M È Kapovich, Conformal structures with Fuchsian holonomy, Dokl. Akad. Nauk SSSR 301 (1988) 23

[22] M Kapovich, On monodromy of complex projective structures, Invent. Math. 119 (1995) 243

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

[24] L Keen, C Series, Continuity of convex hull boundaries, Pacific J. Math. 168 (1995) 183

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

[26] Y N Minsky, Harmonic maps, length, and energy in Teichmüller space, J. Differential Geom. 35 (1992) 151

[27] Y N Minsky, On dynamics of $\operatorname{Out}(F_n)$ on $\operatorname{PSL}_2 (\mathbb{C})$ characters, Israel J. Math. 193 (2013) 47

[28] R C Penner, J L Harer, Combinatorics of train tracks, Annals of Mathematics Studies 125, Princeton Univ. Press (1992)

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

[30] W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)

[31] W P Thurston, Three-dimensional geometry and topology, I, Princeton Mathematical Series 35, Princeton Univ. Press (1997)