Geometry & Topology

Asymptoticity of grafting and Teichmüller rays

Subhojoy Gupta

Full-text: Open access


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 (2π–) 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.

Article information

Geom. Topol., Volume 18, Number 4 (2014), 2127-2188.

Received: 23 December 2012
Accepted: 3 February 2014
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F60: Teichmüller theory [See also 32G15]
Secondary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 57M50: Geometric structures on low-dimensional manifolds

grafting rays Teichmüller rays


Gupta, Subhojoy. Asymptoticity of grafting and Teichmüller rays. Geom. Topol. 18 (2014), no. 4, 2127--2188. doi:10.2140/gt.2014.18.2127.

Export citation


  • L V Ahlfors, Lectures on quasiconformal mappings, 2nd edition, University Lecture Series 38, Amer. Math. Soc. (2006)
  • A Beurling, L Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956) 125–142
  • C J Bishop, Bi-Lipschitz approximations of quasiconformal maps, Ann. Acad. Sci. Fenn. Math. 27 (2002) 97–108
  • S A Bleiler, A J Casson, Automorphisms of surfaces after Nielsen and Thurston, London Math. Soc. Student Texts 9, Cambridge Univ. Press (1988)
  • F Bonahon, Shearing hyperbolic surfaces, bending pleated surfaces and Thurston's symplectic form, Ann. Fac. Sci. Toulouse Math. 5 (1996) 233–297
  • R D Canary, D B A Epstein, P Green, Notes on notes of Thurston, from: “Analytical and geometric aspects of hyperbolic space”, (D B A Epstein, editor), London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 3–92
  • Y-E Choi, D Dumas, K Rafi, Grafting rays fellow travel Teichmüller geodesics, Int. Math. Res. Not. 2012 (2012) 2445–2492
  • R Díaz, I Kim, Asymptotic behavior of grafting rays, Geom. Dedicata 158 (2012) 267–281
  • D Dumas, The Schwarzian derivative and measured laminations on Riemann surfaces, Duke Math. J. 140 (2007) 203–243
  • D Dumas, Complex projective structures, from: “Handbook of Teichmüller theory, Vol. II”, (A Papadopoulos, editor), IRMA Lect. Math. Theor. Phys. 13, Eur. Math. Soc., Zürich (2009) 455–508
  • D Dumas, M Wolf, Projective structures, grafting and measured laminations, Geom. Topol. 12 (2008) 351–386
  • 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”, (R D Canary, D Epstein, A Marden, editors), London Math. Soc. Lecture Note Ser. 328, Cambridge Univ. Press, Cambridge (2006) 117–266
  • G Faltings, Real projective structures on Riemann surfaces, Compositio Math. 48 (1983) 223–269
  • A Fathi, F Laudenbach, V Poénaru, Thurston's work on surfaces, Mathematical Notes 48, Princeton Univ. Press (2012)
  • W M Goldman, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987) 297–326
  • S Gupta, Asymptoticity of grafting and Teichmüller rays, II, to appear in Geom. Dedicata.
  • S Gupta, On the asymptotic behavior of complex earthquakes and Teichmüller disks, to appear in AMS Contemp. Math. Proceedings
  • J H Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics, Vol. 1, Matrix Editions, Ithaca, NY (2006)
  • J H Hubbard, H Masur, Quadratic differentials and foliations, Acta Math. 142 (1979) 221–274
  • Y Imayoshi, M Taniguchi, An introduction to Teichmüller spaces, Springer, Tokyo (1992)
  • N V Ivanov, Isometries of Teichmüller spaces from the point of view of Mostow rigidity, from: “Topology, ergodic theory, real algebraic geometry”, (V Turaev, A Vershik, editors), Amer. Math. Soc. Transl. Ser. 2 202, Amer. Math. Soc. (2001) 131–149
  • Y Kamishima, S P Tan, Deformation spaces on geometric structures, from: “Aspects of low-dimensional manifolds”, (Y Matsumoto, S Morita, editors), Adv. Stud. Pure Math. 20, Kinokuniya, Tokyo (1992) 263–299
  • M Kapovich, Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Birkhäuser, Boston (2009)
  • R S Kulkarni, U Pinkall, A canonical metric for Möbius structures and its applications, Math. Z. 216 (1994) 89–129
  • A Lenzhen, H Masur, Criteria for the divergence of pairs of Teichmüller geodesics, Geom. Dedicata 144 (2010) 191–210
  • H Masur, Uniquely ergodic quadratic differentials, Comment. Math. Helv. 55 (1980) 255–266
  • H Masur, Dense geodesics in moduli space, from: “Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference”, (I Kra, B Maskit, editors), Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 417–438
  • H Masur, Interval exchange transformations and measured foliations, Ann. of Math. 115 (1982) 169–200
  • C T McMullen, Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998) 283–320
  • C Pommerenke, Boundary behaviour of conformal maps, Grundl. Math. Wissen. 299, Springer, Berlin (1992)
  • K P Scannell, M Wolf, The grafting map of Teichmüller space, J. Amer. Math. Soc. 15 (2002) 893–927
  • K Strebel, Quadratic differentials, Ergeb. Math. Grenzgeb. 5, Springer, Berlin (1984)
  • H Tanigawa, Grafting, harmonic maps and projective structures on surfaces, J. Differential Geom. 47 (1997) 399–419
  • W P Thurston, Minimal stretch maps between hyperbolic surfaces
  • W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url {\unhbox0