Hyperbolicity in Teichmüller space

Rafi, Kasra¹

¹ Department of Mathematics, University of Toronto, Room 6290, 40 George Street, Toronto ON M5S 2E4, Canada

Geometry & topology, Tome 18 (2014) no. 5, pp. 3025-3053.

Résumé

We give an inductive description of a Teichmüller geodesic, that is, we show that there is a sense in which a Teichmüller geodesic is assembled from Teichmüller geodesics in smaller subsurfaces. We then apply this description to answer various questions about the geometry of Teichmüller space, obtaining several applications: (1) We show that Teichmüller geodesics do not backtrack in any subsurface. (2) We show that a Teichmüller geodesic segment whose endpoints are in the thick part has the fellow traveling property and that this fails when the endpoints are not necessarily in the thick part. (3) We prove a thin-triangle property for Teichmüller geodesics. Namely, we show that if an edge of a Teichmüller geodesic triangle passes through the thick part, then it is close to one of the other edges.

Détail

DOI : 10.2140/gt.2014.18.3025

Classification : 30F60, 32Q05
Keywords: Teichmüller space, geodesics, fellow traveling, subsurface projection, curve complex

Bibliographie

[1] L Bers, An extremal problem for quasiconformal mappings and a theorem by Thurston, Acta Math. 141 (1978) 73

[2] Y E Choi, K Rafi, Comparison between Teichmüller and Lipschitz metrics, J. Lond. Math. Soc. 76 (2007) 739

[3] Y E Choi, K Rafi, C Series, Lines of minima and Teichmüller geodesics, Geom. Funct. Anal. 18 (2008) 698

[4] F P Gardiner, N Lakic, Quasiconformal Teichmüller theory, Mathematical Surveys and Monographs 76, Amer. Math. Soc. (2000)

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

[6] S P Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980) 23

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

[8] A Lenzhen, K Rafi, Length of a curve is quasi-convex along a Teichmüller geodesic, J. Differential Geom. 88 (2011) 267

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

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

[11] H A Masur, Y N Minsky, Geometry of the complex of curves, II: Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902

[12] H Masur, L Mosher, S Schleimer, On train-track splitting sequences, Duke Math. J. 161 (2012) 1613

[13] H A Masur, M Wolf, Teichmüller space is not Gromov hyperbolic, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995) 259

[14] Y N Minsky, Extremal length estimates and product regions in Teichmüller space, Duke Math. J. 83 (1996) 249

[15] K Rafi, A characterization of short curves of a Teichmüller geodesic, Geom. Topol. 9 (2005) 179

[16] K Rafi, A combinatorial model for the Teichmüller metric, Geom. Funct. Anal. 17 (2007) 936

[17] K Rafi, Thick-thin decomposition for quadratic differentials, Math. Res. Lett. 14 (2007) 333

[18] K Rafi, S Schleimer, Covers and the curve complex, Geom. Topol. 13 (2009) 2141

[19] O Teichmüller, Extremale quasikonforme Abbildungen und quadratische Differentiale, Abh. Preuss. Akad. Wiss. Math.-Nat. Kl. 1939 (1940) 197

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

[21] W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988) 417

[22] W A Veech, The Teichmüller geodesic flow, Ann. of Math. 124 (1986) 441