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Normalizer, divergence type, and Patterson measure for discrete groups of the Gromov hyperbolic space. (English) Zbl 1528.20072

Summary: For a non-elementary discrete isometry group \(G\) of divergence type acting on a proper geodesic \(delta\)-hyperbolic space, we prove that its Patterson measure is quasi-invariant under the normalizer of \(G\). As applications of this result, we have: (1) under a minor assumption, such a discrete group \(G\) admits no proper conjugation, that is, if the conjugate of \(G\) is contained in \(G\), then it coincides with \(G\); (2) the critical exponent of any non-elementary normal subgroup of \(G\) is strictly greater than the half of that for \(G\).

MSC:

20F65 Geometric group theory
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20F67 Hyperbolic groups and nonpositively curved groups
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
20E08 Groups acting on trees
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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