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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: 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.

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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
MR Author ID: 199039

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