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Transactions of the American Mathematical Society

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Limit sets of discrete groups of isometries
of exotic hyperbolic spaces

Authors: Kevin Corlette and Alessandra Iozzi
Journal: Trans. Amer. Math. Soc. 351 (1999), 1507-1530
MSC (1991): Primary 58F11; Secondary 53C35, 58F17
MathSciNet review: 1458321
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [Ad] Anderson, M., The Dirichlet problem at infinity, J. Diff. Geom. 18 (1983), 701-721. MR 85m:58178
  • [As] Anosov, D. V., Geodesic flows on closed Riemannian manifolds with negative curvature, Proceedings of the Steklov Institute of Mathematics, vol. 90, A. M. S., Providence, RI, 1969. MR 39:3527
  • [Au] Auslander, L., Bieberbach's theorem on space groups and discrete uniform subgroups of Lie groups, II, Amer. J. Math. 83 (1961), 276-280. MR 23:A292
  • [Bi-J] Bishop, C. and Jones, P., Hausdorff dimension and Kleinian groups, preprint.
  • [Bu] Bourdon, M., Structure conforme au bord et flot géodésique d' un CAT(-1)-espace, L' Enseignement Math. 41 (1995), 63-102. MR 96f:58120
  • [Bw] Bowditch, B. H., Geometrical finiteness with variable curvature, J. Func. An. 113 (1993), 245-317. MR 94e:57016
  • [C] Corlette, K., Hausdorff dimensions of limit sets I, Invent. Math. 102 (1990), 521-542. MR 91k:58067
  • [E-O] Eberlein, P. and O'Neill, B., Visibility manifolds, Pac. J. Math. 46 (1973), 45-109. MR 49:1421
  • [Go] Goldman, W., A user's guide to complex hyperbolic geometry, Oxford Math. Monographs (to appear).
  • [Gr] 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). CMP 18:09
  • [He1] Helgason, S., Groups and geometric analysis, Academic Press, New York, 1984. MR 86c:22017
  • [He2] -, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York, 1989. MR 80k:53081
  • [Ho1] Hopf, E., Ergodentheorie, Ergebnisse der Mathematik, Band 5, no.2, Springer-Verlag, 1937.
  • [Ho2] -, Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bull. Am. Math. Soc. 77 (1971), 863-877. MR 44:1789
  • [K] Kaimanovich, V. A., Invariant measures for the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. Henri Poincaré, Physique Théorique 53 (4) (1990), 361-393. MR 92b:58176
  • [Mi] Mitchell, J., On Carnot-Carathéodory metrics, J. Diff. Geom. 21 (1985), 35-45. MR 87d:53086
  • [Mo] Mostow, G., Strong rigidity of locally symmetric spaces, vol. 78, Ann. Math. Stud., Princeton University Press, Princeton, 1978. MR 52:5874
  • [Pn1] Pansu, P., Thèse.
  • [Pn2] -, Une inégalité isopérimétrique sur le groups de Heisenberg, C. R. Acad. Sci. Paris Sér. I295 (1982), 127-130. MR 85b:53044
  • [Pn3] -, Metrique de Carnot-Carathéodory et quasi-isométries des espaces symmetrique de rang un, Ann. Math. 129 (1989), 1-60. MR 90e:53058
  • [Pt] Patterson, S. J., The limit set of a Fuchsian group, Acta Math. 136 (1976), 241-273. MR 56:8841
  • [St] Strichartz, R., Sub-Riemannian geometry, J. Diff. Geom. 24 (1980), 221-263. MR 88b:53055
  • [Su1] Sullivan, D., The density at infinity of a discrete group of hyperbolic motions, Publ. I.H.E.S. 50 (1979), 171-202. MR 81b:58031
  • [Su2] -, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984), 259-277. MR 86c:58093
  • [Su3] -, Related aspects of positivity in Riemannian manifolds, J. Diff. Geom. 25 (1987), 327-351. MR 88d:58132
  • [Y] Yue, C., The ergodic theory of discrete isometry groups of manifolds of variable negative curvature, preprint.
  • [W] Wang, H., Discrete subgroups of solvable Lie groups I, Ann. of Math. 64(1) (1956), 1-19. MR 17:1224c

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Additional Information

Kevin Corlette
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637

Alessandra Iozzi
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

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.
Article copyright: © Copyright 1999 American Mathematical Society