The geometry of Teichmüller space via geodesic currents. (English) Zbl 0653.32022

Let S be a compact orientable surface of genus \(g\geq 2\). Denote by \({\mathcal T}(S)\) its Teichmüller space, i.e., the space of isotopy classes of hyperbolic metrics on S. The author provides a very natural compactification of the Teichmüller space \({\mathcal T}(S)\), which concludingly turns out to coincide with W. Thurston’s compactification via projective measured laminations [cf. papers of A. Fathi, F. Laudenbach and V. Poenaru in the book “Travaux de Thurston sur les surfaces” (1979; Zbl 0406.00016); e.g. F. Laudenbach, ibid., 209- 224 (1979; Zbl 0446.57018), A. Fathi and F. Laudenbach, ibid., 139-150 (1979; Zbl 0446.57015), and V. Poenaru, ibid., 5-20 (1979; Zbl 0446.57005)]. The authors construction is based upon the notion of geodesic currents, which has been introduced by himself in an earlier work [cf. the author, Ann. Math. 124, 71-158 (1986)]. The geodesic currents, i.e., the \(\pi_ 1(S)\)-invariant positive measures on the space \(G(\tilde S)\) of (unoriented) geodesics on the universal covering \(\tilde S\) of S, are shown to form a complete uniform space \({\mathcal C}(S)\), whose projectivization \({\mathcal P}{\mathcal C}(S):=({\mathcal C}(S)- 0)/{\mathbb{R}}^+\) is compact. It is then proved that \({\mathcal T}(S)\) admits a proper topological embedding into \({\mathcal C}(S)\), whose image is asymptotic to Thurston’s space \({\mathcal M}{\mathcal L}(S)\) of measured laminations. This provides a compactification of \({\mathcal T}(S)\) (within \({\mathcal P}{\mathcal C}(S))\) by \({\mathcal P}{\mathcal M}{\mathcal L}(S)\), so to speak a “unified” version of Thurston’s approach. Another advantage of the author’s construction is that it gives a representation of Teichmüller space \({\mathcal T}(S)\) as a submanifold of an infinite-dimensional analog of the hyperbolic n-space \({\mathbb{H}}^ n\). Then the metric on \({\mathcal T}(S)\) induced by the hyperbolic metric on \({\mathbb{H}}^ n\) is equal (up to a constant factor) to the celebrated Petersson-Weil metric.
Reviewer: W.Kleinert


32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32J05 Compactification of analytic spaces
58A25 Currents in global analysis
57R30 Foliations in differential topology; geometric theory
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[1] [A-A] Arnol’d, V.I., Avez, A.: Problèmes ergodiques de la mécanique classique. Gauthier-Villars, 1967 (English translation published by Benjamin, 1968) · Zbl 0149.21704
[2] [B-C] Bleiler, S., Casson, A.: Automorphisms of surfaces after Nielsen and Thurston. To appear in Cambridge University Press · Zbl 0649.57008
[3] [B1] Bonahon, F.: Bouts des variétés hyperboliques de dimension 3. Ann. Math.124, 71-158 (1986) · Zbl 0671.57008 · doi:10.2307/1971388
[4] [B2] Bonahon, F.: Structures géométriques sur les variétés de dimension 3 et applications. Thèse d’Etat, Université d’Orsay, 1985
[5] [Bk] Bourbaki, N.: Eléments de Mathématiques, livre VII (Intégration). Paris: Hermann 1965
[6] [C] Chu, T.: The Weil-Petersson metric in the moduli space. Chin. J. Math.4, 29-51 (1976) · Zbl 0344.32006
[7] [E-O] Eberlein, P., O’Neill, B.: Visibility manifolds. Pac. J. Math.46, 45-109 (1973) · Zbl 0264.53026
[8] [F-L-P] Fathi, A., Laudenbach, F., Poenaru, V.: Travaux de Thurston sur les surfaces. Astérisque no 66-67, Société Mathématique de France, 1979
[9] [F] Floyd, W.: Group completions and limit sets of Kleinian groups. Invent. Math.57, 205-218 (1980) · Zbl 0428.20022 · doi:10.1007/BF01418926
[10] [G1] Gromov, M.: Structures métriques pour les variétés riemanniennes. (Notes by Lafontaine, J. and Pansu, P.). Cedic-Fernand-Nathan, 1981
[11] [G2] Gromov, M.: Hyperbolic manifolds, groups and actions. In: Kra, I., Maskit, B. (eds.), Riemann surfaces and related topics. Proceedings of the 1978 Stony Brook conference, (Ann. Math. Studies, vol. 97, pp. 183-215), Princeton University Press, 1981
[12] [G3] Gromov, M.: Hyperbolic spaces. In: Gersten, S.M. (ed.) Essays in combinatorial group theory. Berlin Heidelberg New York: Springer, 1987
[13] [H-P] Harer, J., Penner, R.C.: Combinatorics of train tracks, Preprint, University of Maryland and University of Southern California, 1984 · Zbl 0765.57001
[14] [H] Horowitz, R.: Characters of free groups represented in the two-dimensional linear group. Commun. Pure Appl. Math.25, 635-649 (1972) · Zbl 1184.20009 · doi:10.1002/cpa.3160250602
[15] [J] Jørgensen, T.: Traces in 2-generator subgroups of ?. Proc. Am. Math. Soc.84, 339-343 (1982) · Zbl 0498.20035
[16] [K] Kerckhoff, S.P.: The Nielsen realization theorem. Ann. Math.117, 235-265 (1983) · Zbl 0528.57008 · doi:10.2307/2007076
[17] [N] Nielsen, J.: Untersuchung zur Topologie der geschlossenen zweiseitigen Flächen, I, II and III. Acta Math.50, 189-358 (1927);53, 1-76 (1929);58, 87-167 (1931) · JFM 53.0545.12 · doi:10.1007/BF02421324
[18] [S] Sigmund, K.: On dynamical systems with the specification property. Trans. Am. Math. Soc.190, 285-299 (1974) · Zbl 0286.28010 · doi:10.1090/S0002-9947-1974-0352411-X
[19] [T1] Thurston, W.P.: On the geometry and dynamics of diffeomorphisms of surfaces I. Unpublished article, Princeton University, 1975
[20] [T2] Thurston, W.P.: The topology and geometry of 3-manifolds, lecture notes. Princeton University, 1976-79
[21] [Tr] Tromba, A.J.: On a natural affine connection on the space of almost complex structures and the curvature of Teichmüller space with respect to its Weil-Petersson metric. Manuscr. Math.56, 456-497 (1986) · Zbl 0606.32014 · doi:10.1007/BF01168506
[22] [Wa] Walter, P.: An introduction to ergodic theory. Graduate texts in Mathematics, vol. 79. Berlin-Heidelberg-New York: Springer 1982
[23] [W1] Wolpert, S.: Non completeness of the Weil-Petersson metric for Teichmüller space. Pac. J. Math.61, 573-577 (1975) · Zbl 0327.32009
[24] [W2] Wolpert, S.: Thurston’s Riemannian metric for Teichmüller space. J. Differ. Geom.23, 143-174 (1986) · Zbl 0592.53037
[25] [W3] Wolpert, S.: Chern forms and the Riemann tensor for the moduli space of curves. Invent. Math.85, 119-145 (1986) · Zbl 0595.32031 · doi:10.1007/BF01388794
[26] [W4] Wolpert, S.: Geodesic length functions and the Nielsen problem. J. Differ. Geom.25, 275-296 (1987) · Zbl 0616.53039
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