The Poisson Boundary of the Mapping Class Group and of Teichmüller Space

Kaimanovich, Vadim A.; Masur, Howard

Publications de l'Institut de recherche mathématiques de Rennes, no. 2 (1994), article no. 4, 69 p.

[Ab] W. Abikoff, Degenerating families of Riemann surfaces, Ann. of Math. 105 (1977), 29-44. | MR | Zbl

[An1] A. Ancona, Negatively curved manifolds, elliptic operators and the Martin boundary, Ann. of Math. 125 (1987), 495-536. | MR | Zbl

An2 ]A. Ancona, Théorie du potentiel sur les graphes et les variétés, Springer Lecture Notes in Math. 1427 (1990), 4-112. | MR | Zbl

[AS] M. T. Anderson, R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. 121 (1985), 429-461. | MR | Zbl

[Ba] W. Ballmann, On the Dirichlet problem at infinity for manifolds of nonpositive curvature, Forum Math. 1 (1989), 201-213. | MR | Zbl

[Be] L. Bers, Quasiconformal mappings and Teichmüller's theorem, Analytic Functions (R. Nevanlinna et al., eds.), Princeton University Press, Princenton, New Jersey, 1960, pp.89-119 | MR | Zbl

[BGS] W. Ballmann, M. Gromov, V. Schroeder, Manifolds of Nonpositive Curvature, Birkhäuser, Basel, 1985. | MR | Zbl

[Bi] C. J. Bishop, A characterization of Poissonian domains, Ark. Mat. 29 (1991), 1-24. | MR | Zbl

[BL1] W. Ballmann, F. Ledrappier, The Poisson boundary for rank 1 manifolds and their cocompact lattices, Forum Math. 6 (1994), 301-313. | MR | Zbl

[BL2] W. Ballmann, F. Ledrappier, Discretization of positive harmonic functions on Riemannian manifolds and Martin boundary, preprint (1993). | MR | Zbl

[CB] A. Casson, S. Bleiler, Automorphisms of Surfaces after Nielsen and Thurston, London Math. Soc. Student Texts, vol. 9, Cambridge Univ. Press, 1988. | MR | Zbl

[CDP] M. Coornaert, T. Delzant, A. Papadopoulos, Geometrie et théorie des groupes, Lecture Notes in Math., vol. 1441, Springer-Verlag, Berlin, 1990. | MR | Zbl

[CFS] I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai, Ergodic Theory, Springer-Verlag, Berlin, 1982. | MR | Zbl

[CS] D. I. Cartwright, P. M. Soardi, Convergence to ends for random walks on the automorphism group of a tree, Proc Amer. Math. Soc. 107 (1989), 817-823. | MR | Zbl

[De1] Y. Derriennic, Quelques applications du théorème ergodique sous-additif, Astérisque 74 (1980), 183-201. | Numdam | MR | Zbl

[De2] Y. Derriennic, Entropie, théorèmes limites et marches aléatoires, Springer Lecture Notes in Math. 1210 (1986), 241-284. | MR | Zbl

[FK] H. Farkas, I. Kra, Riemann Surfaces, Springer-Verlag, New York, 1980. | MR | Zbl

[FLP] A. Fathi, F. Laudenbach, V. Poenaru, Travaux de Thurston sur les surfaces, Astérisque, vol. 66-67, 1979. | Numdam | MR | Zbl

[Fo] S. R. Foguel, Harris operators, Israel J. Math. 33 (1979), 281-309. | MR | Zbl

[Fu1] H. Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. 77 (1963), 335-386. | MR | Zbl

[Fu2] H. Furstenberg, Poisson boundaries and envelopes of discrete groups, Bull. Amer. Math. Soc. 73 (1967), 350-356. | MR | Zbl

[Fu3] H. Furstenberg, Random walks and discrete subgroups of Lie groups, Adv. Probab. Related Topics, vol. 1, Dekker, New York, 1971, pp. 3-63. | MR | Zbl

[Ga] F. Gardiner, Teichmüller Theory and Quadratic Differentials, Wiley, New York, 1987. | MR | Zbl

[GH] E. Ghys, P. de la Harpe (eds.), Sur les groupes hyperbolique d'après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser, Basel, 1990. | MR | Zbl

[Gr] M. Gromov, Hyperbolic groups, Essays in Group Theory (S. M. Gersten, ed.), MSRI Publ., vol. 8, Springer, New York, 1987, pp. 75-263. | MR | Zbl

[GR] Y. Guivarc'H, A. Raugi, Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Z. Wahr. 69 (1985), 187-242. | MR | Zbl

[Ha] W. J. Harvey, Geometric structure of surface mapping class groups, Homological Group Theory (C. T. C. Wall, ed.), London Math. Soc. Lecture Note Series, vol. 36, Cambridge Univ. Press, 1979, pp. 255-269. | MR | Zbl

[HM] J. Hubbard, H. Masur, Quadratic differentials and foliations,Acta Math. 142 (1979), 221-274. | MR | Zbl

[Iv1] N. Ivanov, Rank of Teichmüller modular groups, Mat. Zametki 44 (1988), 636-644. | MR | Zbl

[Iv2] N. Ivanov, Automorphisms of Teichmüller modular groups, Lecture Notes in Math. 1346 (1988), 199-270. | MR | Zbl

[Ka1] V. A. Kaimanovich, An entropy criterion for maximality of the boundary of random walks on discrete groups, Soviet Math. Dokl. 31 (1985), 193-197. | MR | Zbl

[Ka2] V. A. Kaimanovich, Brownian motion and harmonic functions on covering manifolds. An entropy approach, Soviet Math. Dokl. 33 (1986), 812-816. | MR | Zbl

[Ka3] V. A. Kaimanovich, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. H. Poincaré, Physique théorique 53 (1990), 361-393. | Numdam | MR | Zbl

[Ka4] V. A. Kaimanovich Discretization of bounded harmonic functions on Riemannian manifolds and entropy, Proceedings of the International Conference on Potential Theory (Nagoya, 1990) (M. Kishi, ed.), de Gruyter, Berlin, 1992, pp.213-223. | MR | Zbl

[Ka5] V. A. Kaimanovich, Measure-theoretic boundaries of Markov chains, 0-2 laws and entropy, Proceedings of the Conference on Harmonic Analysis and Discrete Potential Theory (Frascati, 1991) (M. A. Picardello, ed.), Plenum, New York, 1992, pp. 145-180. | MR

[Ka6] V. A. Kaimanovich, Bi-harmonic functions on groups, C R . Ac. Sci. Paris 314 (1992), 259-264. | MR | Zbl

[Ka7] V. A. Kaimanovich The Poisson boundary of hyperbolic groups,, C R . Ac. Sci. Paris 318 (1994), 59-64. | MR | Zbl

[Ka8] V. A. Kaimanovich, Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. reine und. angew. Math. 455 (1994), 57-103. | MR | Zbl

[Ka9] V. A. Kaimanovich, The Poisson boundary of covering Markov operators, Israel J. Math (1995). | MR | Zbl

[Ka10] V. A. Kaimanovich, A Poisson formula for groups with hyperbolic properties, preprint (1994). | MR | Zbl

[Ka11] V. A. Kaimanovich, An introduction to boundary theory of invariant Markov operators, Proceedings of the Conference on Algebraic and Number Theoretic Aspects of Ergodic Theory (Warwick, 1994) (to appear).

[Kan] M. Kanai, Rough isometries and combinatorial approximation of geometries on non-compact Riemannian manifolds, J. Math. Soc. Japan 37 (1985), 391-413. | MR | Zbl

[Ke1] S. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980), 23-41. | MR | Zbl

[Ke2] S. Kerckhoff, The Nielsen realization problem, Ann. of Math. 117 (1983), 235-265. | MR | Zbl

[Ke3] S. Kerckhoff, Simplicial systems for interval exchange maps and measured foliations, Erg. Th. Dyn. Syst. 5 (1985), 257-271. | MR | Zbl

[Kr] I. Kra, Horocyclic coordinates for Riemann surfaces and moduli space, Jour. Amer. Math. Soc. 3 (1990), 499-578. | MR | Zbl

[Kre] U. Krengel, Ergodic Theorems, de Gruyter, Berlin, 1985. | MR | Zbl

[KV] V. A. Kaimanovich, A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Prob. 11 (1983), 457-490. | MR | Zbl

[Le1] F. Ledrappier, Poisson boundaries of discrete groups of matrices, Israel J. Math. 50 (1985), 319-336. | MR | Zbl

[Le2] F. Ledrappier, Applications of dynamics to compact manifokds of negative curvature, Proc. International Cong. Math. (Zürich, 1994) (to appear). | MR | Zbl

[LMR] A. Lubotzky, S. Mozes, M. S. Raghunathan, Cyclic subgroups of exponential growth and metrics on discrete groups, C.R. Ac. Sci. Paris 317 (1993), 735-740. | MR | Zbl

[LS] T. Lyons, D. Sullivan, Function theory, random paths, and covering spaces, J. Diff. Geom. 19 (1984), 299-323. | MR | Zbl

[Ma1] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. 115 (1982), 169-200. | MR | Zbl

[Ma2] H. Masur, Two boundaries of Teichmuller space, Duke J. Math 49 (1982), 183-190. | MR | Zbl

[Ma3]H. Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke J. Math. 66 (1992), 387-442. | MR | Zbl

[Ma4] H. Masur, Random walks on Teichmüller space and the mapping class group, submitted (1994). | MR | Zbl

[Mar] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, Berlin, 1991. | MR | Zbl

[Mc] J. Mccarthy, A Tits alternative for subgroups of surface mapping class groups, Trans. Amer. Math. Soc. 291 (1985), 583-612. | MR | Zbl

[Mi] Y. Minsky, Extremal length estimates and product regions in Teichmüller space, preprint (1993). | MR | Zbl

[Mil] J. Milnor, A note on curvature and funndamental group, J. Diff. Geom. 2 (1968), 1-7. | MR | Zbl

[Mo] G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Ann. of Math. Studies, vol. 78, Princeton University Press, Princeton New Jersey, 1973. | MR | Zbl

[MP] T. S. Mountford, S. C. Port, Representations of bounded harmonic functions, Ark. Mat. 29 (1991), 107-126. | MR | Zbl

[Mu] D. Mumford, The structure of the moduli space of curves and Abelian varieties, Proc. International Cong. Math. (Nice, 1990), vol. 1 pp. 457-465 | MR | Zbl

[MW] H. Masur, M. Wolf, Teichmüller space is not Gromov hyperbolic, Annales Academiae Scientarum Fennicae (to appear). | Zbl

[Pa] S. J. Patterson, Lectures on measures on limit sets of Kleinian groups. Analytical and Geometric Aspects of Hyperbolic Space (D. B. A. Epstein, ed.), London Math. Soc. Lecture Note Series, vol. 111 , Cambridge Univ. Press, 1987, pp. 281-323. | MR | Zbl

[Pe] R. Penner, Combinatorics of Train Tracks, Ann. of Math. Studies, vol. 125, Princeton University Press, Princeton New Jersey, 1992. | MR | Zbl

[Ra] A. Raugi, Fonctions harmoniques sur les groupes localement compacts à base dénombrable, Bull. Soc. Math. France. Mémoire 54 (1977), 5-118. | Numdam | MR | Zbl

[Re] M. Rees, An alternative approach to the ergodic theory of measured foliations on surfaces, Ergod. Th. Dynam. Sys. 1 (1981), 461-488. | MR | Zbl

[Rev] D. Revuz, Markov Chains, 2nd revised ed., North-Holland, Amsterdam, 1984 | MR | Zbl

[Ro] J. Rosenblatt, Ergodic and mixing random walks on locally compact groups, Math. Ann. 257 (1981), 31-42. | MR | Zbl

[St] K. Strebel, Quadratic Differentials, Springer-Verlag, Berlin, 1984. | MR | Zbl

[Su] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Publ. Math. IHES 50 (1979), 171-202. | Numdam | MR | Zbl

[Th] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 18 (1988), 417-431. | MR | Zbl

[Ve] W. Veech, The Teichmuller geodesic flow, Ann. of Math. 124 (1986), 441-530. | MR | Zbl

[Wol] W. Woess, Boundaries of random walks on graphs and groups with infinitely many ends, Israel J. Math. 68 (1989), 271-301. | MR | Zbl

[Wo2]W. Woess, Fixed sets and free subgroups of groups acting on metric spaces, Math. Z. 214 (1993), 425-440. | MR | Zbl

[Zi] R. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Basel, 1984. | MR | Zbl