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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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