Geometry & Topology

Projective structures, grafting and measured laminations

David Dumas and Michael Wolf

Full-text: Open access


We show that grafting any fixed hyperbolic surface defines a homeomorphism from the space of measured laminations to Teichmüller space, complementing a result of Scannell–Wolf on grafting by a fixed lamination. This result is used to study the relationship between the complex-analytic and geometric coordinate systems for the space of complex projective (1) structures on a surface.

We also study the rays in Teichmüller space associated to the grafting coordinates, obtaining estimates for extremal and hyperbolic length functions and their derivatives along these grafting rays.

Article information

Geom. Topol., Volume 12, Number 1 (2008), 351-386.

Received: 23 April 2007
Accepted: 20 August 2007
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: 30F10: Compact Riemann surfaces and uniformization [See also 14H15, 32G15] 30F40: Kleinian groups [See also 20H10] 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 57M50: Geometric structures on low-dimensional manifolds

projective structures grafting measured laminations


Dumas, David; Wolf, Michael. Projective structures, grafting and measured laminations. Geom. Topol. 12 (2008), no. 1, 351--386. doi:10.2140/gt.2008.12.351.

Export citation


  • L Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960) 94–97
  • L Bers, Correction to “Spaces of Riemann surfaces as bounded domains”, Bull. Amer. Math. Soc. 67 (1961) 465–466
  • F Bonahon, Earthquakes on Riemann surfaces and on measured geodesic laminations, Trans. Amer. Math. Soc. 330 (1992) 69–95
  • F Bonahon, Shearing hyperbolic surfaces, bending pleated surfaces and Thurston's symplectic form, Ann. Fac. Sci. Toulouse Math. $(6)$ 5 (1996) 233–297
  • F Bonahon, Geodesic laminations with transverse Hölder distributions, Ann. Sci. École Norm. Sup. $(4)$ 30 (1997) 205–240
  • F Bonahon, Transverse Hölder distributions for geodesic laminations, Topology 36 (1997) 103–122
  • F Bonahon, Variations of the boundary geometry of $3$–dimensional hyperbolic convex cores, J. Differential Geom. 50 (1998) 1–24
  • F Bonahon, Kleinian groups which are almost Fuchsian, J. Reine Angew. Math. 587 (2005) 1–15
  • F Bonahon, J-P Otal, Laminations measurées de plissage des variétés hyperboliques de dimension 3, Ann. of Math. $(2)$ 160 (2004) 1013–1055
  • R Díaz, I Kim, Asymptotic behavior of grafting rays (2007) Preprint
  • D Dumas, Grafting, pruning, and the antipodal map on measured laminations, J. Differential Geom. 74 (2006) 93–118 Erratum, J. Differential Geom. 77 (2007) 175–176
  • D Dumas, The Schwarzian derivative and measured laminations on Riemann surfaces, Duke Math. J. 140 (2007) 203–243
  • D B A Epstein, A Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, from: “Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984)”, London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press, Cambridge (1987) 113–253
  • D B A Epstein, A Marden, V Markovic, Quasiconformal homeomorphisms and the convex hull boundary, Ann. of Math. $(2)$ 159 (2004) 305–336
  • D Gallo, M Kapovich, A Marden, The monodromy groups of Schwarzian equations on closed Riemann surfaces, Ann. of Math. $(2)$ 151 (2000) 625–704
  • F P Gardiner, Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), John Wiley & Sons, New York (1987) A Wiley–Interscience Publication
  • F P Gardiner, H Masur, Extremal length geometry of Teichmüller space, Complex Variables Theory Appl. 16 (1991) 209–237
  • W M Goldman, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987) 297–326
  • D A Hejhal, Monodromy groups and linearly polymorphic functions, Acta Math. 135 (1975) 1–55
  • Y Kamishima, S P Tan, Deformation spaces on geometric structures, from: “Aspects of low-dimensional manifolds”, Adv. Stud. Pure Math. 20, Kinokuniya, Tokyo (1992) 263–299
  • A B Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR 211 (1973) 775–778
  • S P Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980) 23–41
  • S P Kerckhoff, Earthquakes are analytic, Comment. Math. Helv. 60 (1985) 17–30
  • S P Kerckhoff, Simplicial systems for interval exchange maps and measured foliations, Ergodic Theory Dynam. Systems 5 (1985) 257–271
  • F Klein, Vorlesungen über die Hypergeometrische funktion, Springer–Verlag, Berlin (1933)
  • G Levitt, Feuilletages des surfaces, Ann. Inst. Fourier $($Grenoble$)$ 32 (1982) x, 179–217
  • B Maskit, On a class of Kleinian groups, Ann. Acad. Sci. Fenn. Ser. A I No. 442 (1969) 8
  • H Masur, Interval exchange transformations and measured foliations, Ann. of Math. $(2)$ 115 (1982) 169–200
  • C T McMullen, Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998) 283–320
  • S Nag, The complex analytic theory of Teichmüller spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York (1988) A Wiley-Interscience Publication.
  • J-P Otal, Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3, Astérisque (1996) x+159
  • A Papadopoulos, Deux remarques sur la géométrie symplectique de l'espace des feuilletages mesurés sur une surface, Ann. Inst. Fourier $($Grenoble$)$ 36 (1986) 127–141
  • R C Penner, J L Harer, Combinatorics of train tracks, Annals of Mathematics Studies 125, Princeton University Press, Princeton, NJ (1992)
  • I D Platis, Complex symplectic geometry of quasi-Fuchsian space, Geom. Dedicata 87 (2001) 17–34
  • M Rees, An alternative approach to the ergodic theory of measured foliations on surfaces, Ergodic Theory Dynamical Systems 1 (1981) 461–488 (1982)
  • K P Scannell, M Wolf, The grafting map of Teichmüller space, J. Amer. Math. Soc. 15 (2002) 893–927
  • C Series, On Kerckhoff minima and pleating loci for quasi-Fuchsian groups, Geom. Dedicata 88 (2001) 211–237
  • C Series, Limits of quasi-Fuchsian groups with small bending, Duke Math. J. 128 (2005) 285–329
  • C Series, Thurston's bending measure conjecture for once punctured torus groups, from: “Spaces of Kleinian groups”, London Math. Soc. Lecture Note Ser. 329, Cambridge Univ. Press, Cambridge (2006) 75–89
  • Y S özen, F Bonahon, The Weil–Petersson and Thurston symplectic forms, Duke Math. J. 108 (2001) 581–597
  • D Sullivan, W Thurston, Manifolds with canonical coordinate charts: some examples, Enseign. Math. $(2)$ 29 (1983) 15–25
  • H Tanigawa, Grafting, harmonic maps and projective structures on surfaces, J. Differential Geom. 47 (1997) 399–419
  • W P Thurston, Geometry and Topology of Three-Manifolds, Princeton lecture notes (1979) Available at \setbox0\makeatletter\@url {\unhbox0
  • W P Thurston, Earthquakes in two-dimensional hyperbolic geometry, from: “Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984)”, London Math. Soc. Lecture Note Ser. 112, Cambridge Univ. Press, Cambridge (1986) 91–112
  • W P Thurston, Minimal stretch maps between hyperbolic surfaces (1986) Unpublished preprint.
  • A J Tromba, Teichmüller theory in Riemannian geometry, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (1992) Lecture notes prepared by Jochen Denzler.
  • W A Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. $(2)$ 115 (1982) 201–242
  • S Wolpert, An elementary formula for the Fenchel–Nielsen twist, Comment. Math. Helv. 56 (1981) 132–135
  • S Wolpert, The Fenchel–Nielsen deformation, Ann. of Math. $(2)$ 115 (1982) 501–528
  • S Wolpert, On the symplectic geometry of deformations of a hyperbolic surface, Ann. of Math. $(2)$ 117 (1983) 207–234
  • S Wolpert, Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85 (1986) 119–145
  • P G Zograf, L A Takhtadzhyan, On the Liouville equation, accessory parameters and the geometry of Teichmüller space for Riemann surfaces of genus $0$, Mat. Sb. $($N.S.$)$ 132(174) (1987) 147–166