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
Access Full Article
topHow to cite
topKaimanovich, 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
top- [Ab] W. Abikoff, Degenerating families of Riemann surfaces, Ann. of Math.105 (1977), 29-44. Zbl0347.32010MR442293
- [An1] A. Ancona, Negatively curved manifolds, elliptic operators and the Martin boundary, Ann. of Math.125 (1987), 495-536. Zbl0652.31008MR890161
- An2 ] A. Ancona, Théorie du potentiel sur les graphes et les variétés, SpringerLecture Notes in Math.1427 (1990), 4-112. Zbl0719.60074MR1100282
- [AS] M. T. Anderson, R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math.121 (1985), 429-461. Zbl0587.53045MR794369
- [Ba] W. Ballmann, On the Dirichlet problem at infinity for manifolds of nonpositive curvature, Forum Math.1 (1989), 201-213. Zbl0661.53026MR990144
- [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
- [BGS] W. Ballmann, M. Gromov, V. Schroeder, Manifolds of Nonpositive Curvature, Birkhäuser, Basel, 1985. Zbl0591.53001MR823981
- [Bi] C. J. Bishop, A characterization of Poissonian domains, Ark. Mat.29 (1991), 1-24. Zbl0733.31005MR1115072
- [BL1] W. Ballmann, F. Ledrappier, The Poisson boundary for rank 1 manifolds and their cocompact lattices, Forum Math.6 (1994), 301-313. Zbl0801.53028MR1269841
- [BL2] W. Ballmann, F. Ledrappier, Discretization of positive harmonic functions on Riemannian manifolds and Martin boundary, preprint (1993). Zbl0885.53037MR1427756
- [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
- [CDP] M. Coornaert, T. Delzant, A. Papadopoulos, Geometrie et théorie des groupes, Lecture Notes in Math., vol. 1441, Springer-Verlag, Berlin, 1990. Zbl0727.20018MR1075994
- [CFS] I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai, Ergodic Theory, Springer-Verlag, Berlin, 1982. Zbl0493.28007MR832433
- [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
- [De1] Y. Derriennic, Quelques applications du théorème ergodique sous-additif, Astérisque74 (1980), 183-201. Zbl0446.60059MR588163
- [De2] Y. Derriennic, Entropie, théorèmes limites et marches aléatoires, SpringerLecture Notes in Math.1210 (1986), 241-284. Zbl0612.60005MR879010
- [FK] H. Farkas, I. Kra, Riemann Surfaces, Springer-Verlag, New York, 1980. Zbl0764.30001MR583745
- [FLP] A. Fathi, F. Laudenbach, V. Poenaru, Travaux de Thurston sur les surfaces, Astérisque, vol. 66-67, 1979. Zbl0446.57010
- [Fo] S. R. Foguel, Harris operators, Israel J. Math.33 (1979), 281-309. Zbl0434.60012MR571535
- [Fu1] H. Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math.77 (1963), 335-386. Zbl0192.12704MR146298
- [Fu2] H. Furstenberg, Poisson boundaries and envelopes of discrete groups, Bull. Amer. Math. Soc.73 (1967), 350-356. Zbl0184.33105MR210812
- [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
- [Ga] F. Gardiner, Teichmüller Theory and Quadratic Differentials, Wiley, New York, 1987. Zbl0629.30002MR903027
- [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
- [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
- [GR] Y. Guivarc'h, A. Raugi, Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Z. Wahr.69 (1985), 187-242. Zbl0558.60009MR779457
- [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
- [HM] J. Hubbard, H. Masur, Quadratic differentials and foliations,Acta Math.142 (1979), 221-274. Zbl0415.30038MR523212
- [Iv1] N. Ivanov, Rank of Teichmüller modular groups, Mat. Zametki44 (1988), 636-644. Zbl0671.57006MR980584
- [Iv2] N. Ivanov, Automorphisms of Teichmüller modular groups, Lecture Notes in Math.1346 (1988), 199-270. Zbl0657.57004MR970079
- [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
- [Ka2] V. A. Kaimanovich, Brownian motion and harmonic functions on covering manifolds. An entropy approach, Soviet Math. Dokl.33 (1986), 812-816. Zbl0615.60074MR852647
- [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
- [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
- [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
- [Ka6] V. A. Kaimanovich, Bi-harmonic functions on groups, C R . Ac. Sci.Paris314 (1992), 259-264. Zbl0754.60015MR1151710
- [Ka7] V. A. KaimanovichThe Poisson boundary of hyperbolic groups,, C R . Ac. Sci.Paris318 (1994), 59-64. Zbl0792.60006MR1260536
- [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
- [Ka9] V. A. Kaimanovich, The Poisson boundary of covering Markov operators, Israel J. Math (1995). Zbl0843.43001MR1324456
- [Ka10] V. A. Kaimanovich, A Poisson formula for groups with hyperbolic properties, preprint (1994). Zbl0984.60088MR1815698
- [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. Japan37 (1985), 391-413. Zbl0554.53030MR792983
- [Ke1] S. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology19 (1980), 23-41. Zbl0439.30012MR559474
- [Ke2] S. Kerckhoff, The Nielsen realization problem, Ann. of Math.117 (1983), 235-265. Zbl0528.57008MR690845
- [Ke3] S. Kerckhoff, Simplicial systems for interval exchange maps and measured foliations, Erg. Th. Dyn. Syst.5 (1985), 257-271. Zbl0597.58024MR796753
- [Kr] I. Kra, Horocyclic coordinates for Riemann surfaces and moduli space, Jour. Amer. Math. Soc.3 (1990), 499-578. Zbl0714.30040MR1049503
- [Kre] U. Krengel, Ergodic Theorems, de Gruyter, Berlin, 1985. Zbl0575.28009MR797411
- [KV] V. A. Kaimanovich, A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Prob.11 (1983), 457-490. Zbl0641.60009MR704539
- [Le1] F. Ledrappier, Poisson boundaries of discrete groups of matrices, Israel J. Math.50 (1985), 319-336. Zbl0574.60012MR800190
- [Le2] F. Ledrappier, Applications of dynamics to compact manifokds of negative curvature, Proc. International Cong. Math. (Zürich, 1994) (to appear). Zbl0841.53037MR1404020
- [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
- [LS] T. Lyons, D. Sullivan, Function theory, random paths, and covering spaces, J. Diff. Geom.19 (1984), 299-323. Zbl0554.58022MR755228
- [Ma1] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math.115 (1982), 169-200. Zbl0497.28012MR644018
- [Ma2] H. Masur, Two boundaries of Teichmuller space, Duke J. Math49 (1982), 183-190. Zbl0508.30039MR650376
- [Ma3] H. Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke J. Math.66 (1992), 387-442. Zbl0780.30032MR1167101
- [Ma4] H. Masur, Random walks on Teichmüller space and the mapping class group, submitted (1994). Zbl0856.32014MR1383491
- [Mar] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, Berlin, 1991. Zbl0732.22008MR1090825
- [Mc] J. McCarthy, A Tits alternative for subgroups of surface mapping class groups, Trans. Amer. Math. Soc.291 (1985), 583-612. Zbl0579.57006MR800253
- [Mi] Y. Minsky, Extremal length estimates and product regions in Teichmüller space, preprint (1993). Zbl0861.32015MR1390649
- [Mil] J. Milnor, A note on curvature and funndamental group, J. Diff. Geom.2 (1968), 1-7. Zbl0162.25401MR232311
- [Mo] G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Ann. of Math. Studies, vol. 78, Princeton University Press, PrincetonNew Jersey, 1973. Zbl0265.53039MR385004
- [MP] T. S. Mountford, S. C. Port, Representations of bounded harmonic functions, Ark. Mat.29 (1991), 107-126. Zbl0728.31003MR1115078
- [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
- [MW] H. Masur, M. Wolf, Teichmüller space is not Gromov hyperbolic, Annales Academiae Scientarum Fennicae (to appear). Zbl0878.32015
- [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
- [Pe] R. Penner, Combinatorics of Train Tracks, Ann. of Math. Studies, vol. 125, Princeton University Press, PrincetonNew Jersey, 1992. Zbl0765.57001MR1144770
- [Ra] A. Raugi, Fonctions harmoniques sur les groupes localement compacts à base dénombrable, Bull. Soc. Math. France. Mémoire 54 (1977), 5-118. Zbl0389.60003MR517392
- [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
- [Rev] D. Revuz, Markov Chains, 2nd revised ed., North-Holland, Amsterdam, 1984 Zbl0332.60045MR758799
- [Ro] J. Rosenblatt, Ergodic and mixing random walks on locally compact groups, Math. Ann.257 (1981), 31-42. Zbl0451.60011MR630645
- [St] K. Strebel, Quadratic Differentials, Springer-Verlag, Berlin, 1984. Zbl0547.30001MR743423
- [Su] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Publ. Math. IHES50 (1979), 171-202. Zbl0439.30034MR556586
- [Th] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc.18 (1988), 417-431. Zbl0674.57008MR956596
- [Ve] W. Veech, The Teichmuller geodesic flow, Ann. of Math.124 (1986), 441-530. Zbl0658.32016MR866707
- [Wol] W. Woess, Boundaries of random walks on graphs and groups with infinitely many ends, Israel J. Math.68 (1989), 271-301. Zbl0723.60009MR1039474
- [Wo2] W. Woess, Fixed sets and free subgroups of groups acting on metric spaces, Math. Z.214 (1993), 425-440. Zbl0892.54022MR1245204
- [Zi] R. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Basel, 1984. Zbl0571.58015MR776417
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.