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

Vadim A. Kaimanovich; Howard Masur

Publications mathématiques et informatique de Rennes (1994)

  • Issue: 2, page 1-69

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Kaimanovich, Vadim A., and Masur, Howard. "The Poisson Boundary of the Mapping Class Group and of Teichmüller Space." Publications mathématiques et informatique de Rennes (1994): 1-69. <http://eudml.org/doc/274025>.

@article{Kaimanovich1994,
author = {Kaimanovich, Vadim A., Masur, Howard},
journal = {Publications mathématiques et informatique de Rennes},
keywords = {Teichmüller space; hyperbolic space},
language = {eng},
number = {2},
pages = {1-69},
publisher = {Département de Mathématiques et Informatique, Université de Rennes},
title = {The Poisson Boundary of the Mapping Class Group and of Teichmüller Space},
url = {http://eudml.org/doc/274025},
year = {1994},
}

TY - JOUR
AU - Kaimanovich, Vadim A.
AU - Masur, Howard
TI - The Poisson Boundary of the Mapping Class Group and of Teichmüller Space
JO - Publications mathématiques et informatique de Rennes
PY - 1994
PB - Département de Mathématiques et Informatique, Université de Rennes
IS - 2
SP - 1
EP - 69
LA - eng
KW - Teichmüller space; hyperbolic space
UR - http://eudml.org/doc/274025
ER -

References

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  1. [Ab] W. Abikoff, Degenerating families of Riemann surfaces, Ann. of Math.105 (1977), 29-44. Zbl0347.32010MR442293
  2. [An1] A. Ancona, Negatively curved manifolds, elliptic operators and the Martin boundary, Ann. of Math.125 (1987), 495-536. Zbl0652.31008MR890161
  3. An2 ] A. Ancona, Théorie du potentiel sur les graphes et les variétés, SpringerLecture Notes in Math.1427 (1990), 4-112. Zbl0719.60074MR1100282
  4. [AS] M. T. Anderson, R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math.121 (1985), 429-461. Zbl0587.53045MR794369
  5. [Ba] W. Ballmann, On the Dirichlet problem at infinity for manifolds of nonpositive curvature, Forum Math.1 (1989), 201-213. Zbl0661.53026MR990144
  6. [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 Zbl0100.28904MR114898
  7. [BGS] W. Ballmann, M. Gromov, V. Schroeder, Manifolds of Nonpositive Curvature, Birkhäuser, Basel, 1985. Zbl0591.53001MR823981
  8. [Bi] C. J. Bishop, A characterization of Poissonian domains, Ark. Mat.29 (1991), 1-24. Zbl0733.31005MR1115072
  9. [BL1] W. Ballmann, F. Ledrappier, The Poisson boundary for rank 1 manifolds and their cocompact lattices, Forum Math.6 (1994), 301-313. Zbl0801.53028MR1269841
  10. [BL2] W. Ballmann, F. Ledrappier, Discretization of positive harmonic functions on Riemannian manifolds and Martin boundary, preprint (1993). Zbl0885.53037MR1427756
  11. [CB] A. Casson, S. Bleiler, Automorphisms of Surfaces after Nielsen and Thurston, London Math. Soc. Student Texts, vol. 9, Cambridge Univ. Press, 1988. Zbl0649.57008MR964685
  12. [CDP] M. Coornaert, T. Delzant, A. Papadopoulos, Geometrie et théorie des groupes, Lecture Notes in Math., vol. 1441, Springer-Verlag, Berlin, 1990. Zbl0727.20018MR1075994
  13. [CFS] I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai, Ergodic Theory, Springer-Verlag, Berlin, 1982. Zbl0493.28007MR832433
  14. [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. Zbl0682.60059MR984784
  15. [De1] Y. Derriennic, Quelques applications du théorème ergodique sous-additif, Astérisque74 (1980), 183-201. Zbl0446.60059MR588163
  16. [De2] Y. Derriennic, Entropie, théorèmes limites et marches aléatoires, SpringerLecture Notes in Math.1210 (1986), 241-284. Zbl0612.60005MR879010
  17. [FK] H. Farkas, I. Kra, Riemann Surfaces, Springer-Verlag, New York, 1980. Zbl0764.30001MR583745
  18. [FLP] A. Fathi, F. Laudenbach, V. Poenaru, Travaux de Thurston sur les surfaces, Astérisque, vol. 66-67, 1979. Zbl0446.57010
  19. [Fo] S. R. Foguel, Harris operators, Israel J. Math.33 (1979), 281-309. Zbl0434.60012MR571535
  20. [Fu1] H. Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math.77 (1963), 335-386. Zbl0192.12704MR146298
  21. [Fu2] H. Furstenberg, Poisson boundaries and envelopes of discrete groups, Bull. Amer. Math. Soc.73 (1967), 350-356. Zbl0184.33105MR210812
  22. [Fu3] H. Furstenberg, Random walks and discrete subgroups of Lie groups, Adv. Probab. Related Topics, vol. 1, Dekker, New York, 1971, pp. 3-63. Zbl0221.22008MR284569
  23. [Ga] F. Gardiner, Teichmüller Theory and Quadratic Differentials, Wiley, New York, 1987. Zbl0629.30002MR903027
  24. [GH] E. Ghys, P. de la Harpe (eds.), Sur les groupes hyperbolique d'après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser, Basel, 1990. Zbl0731.20025MR1086648
  25. [Gr] M. Gromov, Hyperbolic groups, Essays in Group Theory (S. M. Gersten, ed.), MSRI Publ., vol. 8, Springer, New York, 1987, pp. 75-263. Zbl0634.20015MR919829
  26. [GR] Y. Guivarc'h, A. Raugi, Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Z. Wahr.69 (1985), 187-242. Zbl0558.60009MR779457
  27. [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. Zbl0424.57006MR564431
  28. [HM] J. Hubbard, H. Masur, Quadratic differentials and foliations,Acta Math.142 (1979), 221-274. Zbl0415.30038MR523212
  29. [Iv1] N. Ivanov, Rank of Teichmüller modular groups, Mat. Zametki44 (1988), 636-644. Zbl0671.57006MR980584
  30. [Iv2] N. Ivanov, Automorphisms of Teichmüller modular groups, Lecture Notes in Math.1346 (1988), 199-270. Zbl0657.57004MR970079
  31. [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. Zbl0611.60060MR780288
  32. [Ka2] V. A. Kaimanovich, Brownian motion and harmonic functions on covering manifolds. An entropy approach, Soviet Math. Dokl.33 (1986), 812-816. Zbl0615.60074MR852647
  33. [Ka3] V. A. Kaimanovich, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. H. Poincaré, Physique théorique53 (1990), 361-393. Zbl0725.58026MR1096098
  34. [Ka4] V. A. KaimanovichDiscretization 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. Zbl0768.58054MR1167237
  35. [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. MR1222456
  36. [Ka6] V. A. Kaimanovich, Bi-harmonic functions on groups, C R . Ac. Sci.Paris314 (1992), 259-264. Zbl0754.60015MR1151710
  37. [Ka7] V. A. KaimanovichThe Poisson boundary of hyperbolic groups,, C R . Ac. Sci.Paris318 (1994), 59-64. Zbl0792.60006MR1260536
  38. [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. Zbl0803.58032MR1293874
  39. [Ka9] V. A. Kaimanovich, The Poisson boundary of covering Markov operators, Israel J. Math (1995). Zbl0843.43001MR1324456
  40. [Ka10] V. A. Kaimanovich, A Poisson formula for groups with hyperbolic properties, preprint (1994). Zbl0984.60088MR1815698
  41. [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). 
  42. [Kan] M. Kanai, Rough isometries and combinatorial approximation of geometries on non-compact Riemannian manifolds, J. Math. Soc. Japan37 (1985), 391-413. Zbl0554.53030MR792983
  43. [Ke1] S. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology19 (1980), 23-41. Zbl0439.30012MR559474
  44. [Ke2] S. Kerckhoff, The Nielsen realization problem, Ann. of Math.117 (1983), 235-265. Zbl0528.57008MR690845
  45. [Ke3] S. Kerckhoff, Simplicial systems for interval exchange maps and measured foliations, Erg. Th. Dyn. Syst.5 (1985), 257-271. Zbl0597.58024MR796753
  46. [Kr] I. Kra, Horocyclic coordinates for Riemann surfaces and moduli space, Jour. Amer. Math. Soc.3 (1990), 499-578. Zbl0714.30040MR1049503
  47. [Kre] U. Krengel, Ergodic Theorems, de Gruyter, Berlin, 1985. Zbl0575.28009MR797411
  48. [KV] V. A. Kaimanovich, A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Prob.11 (1983), 457-490. Zbl0641.60009MR704539
  49. [Le1] F. Ledrappier, Poisson boundaries of discrete groups of matrices, Israel J. Math.50 (1985), 319-336. Zbl0574.60012MR800190
  50. [Le2] F. Ledrappier, Applications of dynamics to compact manifokds of negative curvature, Proc. International Cong. Math. (Zürich, 1994) (to appear). Zbl0841.53037MR1404020
  51. [LMR] A. Lubotzky, S. Mozes, M. S. Raghunathan, Cyclic subgroups of exponential growth and metrics on discrete groups, C.R. Ac. Sci.Paris317 (1993), 735-740. Zbl0786.22016MR1244421
  52. [LS] T. Lyons, D. Sullivan, Function theory, random paths, and covering spaces, J. Diff. Geom.19 (1984), 299-323. Zbl0554.58022MR755228
  53. [Ma1] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math.115 (1982), 169-200. Zbl0497.28012MR644018
  54. [Ma2] H. Masur, Two boundaries of Teichmuller space, Duke J. Math49 (1982), 183-190. Zbl0508.30039MR650376
  55. [Ma3] H. Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke J. Math.66 (1992), 387-442. Zbl0780.30032MR1167101
  56. [Ma4] H. Masur, Random walks on Teichmüller space and the mapping class group, submitted (1994). Zbl0856.32014MR1383491
  57. [Mar] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, Berlin, 1991. Zbl0732.22008MR1090825
  58. [Mc] J. McCarthy, A Tits alternative for subgroups of surface mapping class groups, Trans. Amer. Math. Soc.291 (1985), 583-612. Zbl0579.57006MR800253
  59. [Mi] Y. Minsky, Extremal length estimates and product regions in Teichmüller space, preprint (1993). Zbl0861.32015MR1390649
  60. [Mil] J. Milnor, A note on curvature and funndamental group, J. Diff. Geom.2 (1968), 1-7. Zbl0162.25401MR232311
  61. [Mo] G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Ann. of Math. Studies, vol. 78, Princeton University Press, PrincetonNew Jersey, 1973. Zbl0265.53039MR385004
  62. [MP] T. S. Mountford, S. C. Port, Representations of bounded harmonic functions, Ark. Mat.29 (1991), 107-126. Zbl0728.31003MR1115078
  63. [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 Zbl0222.14023MR441983
  64. [MW] H. Masur, M. Wolf, Teichmüller space is not Gromov hyperbolic, Annales Academiae Scientarum Fennicae (to appear). Zbl0878.32015
  65. [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. Zbl0611.30036MR903855
  66. [Pe] R. Penner, Combinatorics of Train Tracks, Ann. of Math. Studies, vol. 125, Princeton University Press, PrincetonNew Jersey, 1992. Zbl0765.57001MR1144770
  67. [Ra] A. Raugi, Fonctions harmoniques sur les groupes localement compacts à base dénombrable, Bull. Soc. Math. France. Mémoire 54 (1977), 5-118. Zbl0389.60003MR517392
  68. [Re] M. Rees, An alternative approach to the ergodic theory of measured foliations on surfaces, Ergod. Th. Dynam. Sys.1 (1981), 461-488. Zbl0539.58018MR662738
  69. [Rev] D. Revuz, Markov Chains, 2nd revised ed., North-Holland, Amsterdam, 1984 Zbl0332.60045MR758799
  70. [Ro] J. Rosenblatt, Ergodic and mixing random walks on locally compact groups, Math. Ann.257 (1981), 31-42. Zbl0451.60011MR630645
  71. [St] K. Strebel, Quadratic Differentials, Springer-Verlag, Berlin, 1984. Zbl0547.30001MR743423
  72. [Su] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Publ. Math. IHES50 (1979), 171-202. Zbl0439.30034MR556586
  73. [Th] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc.18 (1988), 417-431. Zbl0674.57008MR956596
  74. [Ve] W. Veech, The Teichmuller geodesic flow, Ann. of Math.124 (1986), 441-530. Zbl0658.32016MR866707
  75. [Wol] W. Woess, Boundaries of random walks on graphs and groups with infinitely many ends, Israel J. Math.68 (1989), 271-301. Zbl0723.60009MR1039474
  76. [Wo2] W. Woess, Fixed sets and free subgroups of groups acting on metric spaces, Math. Z.214 (1993), 425-440. Zbl0892.54022MR1245204
  77. [Zi] R. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Basel, 1984. Zbl0571.58015MR776417

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