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Limit sets of discrete groups of isometries of exotic hyperbolic spaces. (English) Zbl 0932.37011
Summary: 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’s ergodicity theorem for the geodesic flow with respect to the Bowen-Margulis measure is also proven.

MSC:
 37D05 Dynamical systems with hyperbolic orbits and sets 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 53C35 Differential geometry of symmetric spaces
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References:
 [1] Michael T. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature, J. Differential Geom. 18 (1983), no. 4, 701 – 721 (1984). · Zbl 0541.53036 [2] D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature., Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). Translated from the Russian by S. Feder, American Mathematical Society, Providence, R.I., 1969. · Zbl 0163.43604 [3] Wayman L. Strother, Continuous multi-valued functions, Bol. Soc. Mat. São Paulo 10 (1955), 87 – 120 (1958). · Zbl 0097.38602 [4] Bishop, C. and Jones, P., Hausdorff dimension and Kleinian groups, preprint. · Zbl 0921.30032 [5] Marc Bourdon, Structure conforme au bord et flot géodésique d’un \?\?\?(-1)-espace, Enseign. Math. (2) 41 (1995), no. 1-2, 63 – 102 (French, with English and French summaries). · Zbl 0871.58069 [6] B. H. Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993), no. 2, 245 – 317. · Zbl 0789.57007 [7] Kevin Corlette, Hausdorff dimensions of limit sets. I, Invent. Math. 102 (1990), no. 3, 521 – 541. · Zbl 0744.53030 [8] P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45 – 109. · Zbl 0264.53026 [9] Goldman, W., A user’s guide to complex hyperbolic geometry, Oxford Math. Monographs (to appear). [10] 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 · Zbl 0627.53036 [11] 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. · Zbl 0543.58001 [12] 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. · Zbl 0451.53038 [13] Hopf, E., Ergodentheorie, Ergebnisse der Mathematik, Band 5, no.2, Springer-Verlag, 1937. · JFM 63.0786.07 [14] Eberhard Hopf, Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bull. Amer. Math. Soc. 77 (1971), 863 – 877. · Zbl 0227.53003 [15] 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). · Zbl 0725.58026 [16] John Mitchell, On Carnot-Carathéodory metrics, J. Differential Geom. 21 (1985), no. 1, 35 – 45. · Zbl 0554.53023 [17] G. D. Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. Annals of Mathematics Studies, No. 78. · Zbl 0265.53039 [18] Pansu, P., Thèse. · Zbl 1288.00032 [19] 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). · Zbl 0502.53039 [20] 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). · Zbl 0678.53042 [21] S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), no. 3-4, 241 – 273. · Zbl 0336.30005 [22] Robert S. Strichartz, Sub-Riemannian geometry, J. Differential Geom. 24 (1986), no. 2, 221 – 263. · Zbl 0609.53021 [23] Dennis Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171 – 202. · Zbl 0439.30034 [24] 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. · Zbl 0566.58022 [25] Dennis Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom. 25 (1987), no. 3, 327 – 351. · Zbl 0615.53029 [26] Yue, C., The ergodic theory of discrete isometry groups of manifolds of variable negative curvature, preprint. · Zbl 0864.58047 [27] Wang, H., Discrete subgroups of solvable Lie groups I, Ann. of Math. 64(1) (1956), 1-19. · Zbl 0073.25803
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