Limit sets of discrete groups of isometries of exotic hyperbolic spaces
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- by Kevin Corlette and Alessandra Iozzi PDF
- Trans. Amer. Math. Soc. 351 (1999), 1507-1530 Request permission
Abstract:
Let $\Gamma$ be a geometrically finite discrete group of isometries of hyperbolic space $\mathcal {H}_{\mathbb {F}}^n$, where $\mathbb {F}= \mathbb {R}, \mathbb {C}, \mathbb {H}$ or $\mathbb {O}$ (in which case $n=2$). We prove that the critical exponent of $\Gamma$ equals the Hausdorff dimension of the limit sets $\Lambda (\Gamma )$ and that the smallest eigenvalue of the Laplacian acting on square integrable functions is a quadratic function of either of them (when they are sufficiently large). A generalization of Hopf ergodicity theorem for the geodesic flow with respect to the Bowen-Margulis measure is also proven.References
- Michael T. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature, J. Differential Geom. 18 (1983), no. 4, 701–721 (1984). MR 730923
- D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature. , Proceedings of the Steklov Institute of Mathematics, No. 90 (1967), American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by S. Feder. MR 0242194
- Wayman L. Strother, Continuous multi-valued functions, Bol. Soc. Mat. São Paulo 10 (1955), 87–120 (1958). MR 122961
- Bishop, C. and Jones, P., Hausdorff dimension and Kleinian groups, preprint.
- Marc Bourdon, Structure conforme au bord et flot géodésique d’un $\textrm {CAT}(-1)$-espace, Enseign. Math. (2) 41 (1995), no. 1-2, 63–102 (French, with English and French summaries). MR 1341941
- B. H. Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993), no. 2, 245–317. MR 1218098, DOI 10.1006/jfan.1993.1052
- Kevin Corlette, Hausdorff dimensions of limit sets. I, Invent. Math. 102 (1990), no. 3, 521–541. MR 1074486, DOI 10.1007/BF01233439
- P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45–109. MR 336648, DOI 10.2140/pjm.1973.46.45
- Goldman, W., A user’s guide to complex hyperbolic geometry, Oxford Math. Monographs (to appear).
- Gromov, M., Asymptotic geometry of homogeneous spaces, Conference on Differential geometry on homogeneous spaces (Torino, 1983), Rend. Sem. Mat. Univ. Politec. Torino 1983, Fasc. Spec. 59-60 (1984).
- Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- Hopf, E., Ergodentheorie, Ergebnisse der Mathematik, Band 5, no.2, Springer-Verlag, 1937.
- Eberhard Hopf, Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bull. Amer. Math. Soc. 77 (1971), 863–877. MR 284564, DOI 10.1090/S0002-9904-1971-12799-4
- Vadim A. Kaimanovich, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), no. 4, 361–393 (English, with French summary). Hyperbolic behaviour of dynamical systems (Paris, 1990). MR 1096098
- John Mitchell, On Carnot-Carathéodory metrics, J. Differential Geom. 21 (1985), no. 1, 35–45. MR 806700
- G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. MR 0385004
- Pansu, P., Thèse.
- Pierre Pansu, Une inégalité isopérimétrique sur le groupe de Heisenberg, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 2, 127–130 (French, with English summary). MR 676380
- Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60 (French, with English summary). MR 979599, DOI 10.2307/1971484
- S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), no. 3-4, 241–273. MR 450547, DOI 10.1007/BF02392046
- Robert S. Strichartz, Sub-Riemannian geometry, J. Differential Geom. 24 (1986), no. 2, 221–263. MR 862049
- Dennis Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202. MR 556586, DOI 10.1007/BF02684773
- Dennis Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984), no. 3-4, 259–277. MR 766265, DOI 10.1007/BF02392379
- Dennis Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom. 25 (1987), no. 3, 327–351. MR 882827
- Yue, C., The ergodic theory of discrete isometry groups of manifolds of variable negative curvature, preprint.
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Additional Information
- Kevin Corlette
- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
- Email: kevin@math.uchicago.edu
- Alessandra Iozzi
- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- MR Author ID: 199039
- Email: iozzi@math.umd.edu
- Received by editor(s): February 27, 1995
- Received by editor(s) in revised form: April 15, 1997
- Additional Notes: K. C. received support from a Sloan Foundation Fellowship, an NSF Presidential Young Investigator award, and NSF grant DMS-9203765. A. I. received support from NSF grants DMS 9001959, DMS 9100383 and DMS 8505550.
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 1507-1530
- MSC (1991): Primary 58F11; Secondary 53C35, 58F17
- DOI: https://doi.org/10.1090/S0002-9947-99-02113-3
- MathSciNet review: 1458321