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Patterson-Sullivan measures on the boundary of a hyperbolic space in the sense of Gromov. (Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov.) (French) Zbl 0797.20029
The author investigates the metric theory of the boundary of a discrete group of quasi-isometries of a hyperbolic space (in the sense of Rips- Gromov); note that this is a somewhat more general concept than that of a hyperbolic group. One can define a critical exponent (exponent of convergence) for such a group as \(\lim \sup R^{-1}\log n_ Y(R)\) where \(n_ Y(R)\) denotes the number of elements of the orbit of \(Y\) in a ball of radius \(R\). This is well-defined and the author shows that if the group is infinite this exponent is positive. If the group is finitely generated then it is also finite.
The author shows that there is a measure supported on the limit set of the group (i.e. the accumulation set in the boundary of any orbit in the hyperbolic space on which the group acts). This measure is quasi- conformal under the group with exponent equal to the critical exponent. This construction allows the author to extend the range of the familiar theory of standard hyperbolic groups to this class. In particular he is able to obtain sharp estimates for the growth of \(n_ Y(R)\) and to evaluate the Hausdorff dimension of the limit set of the group. The results are particularly sharp if the hyperbolic space is a tree.

MSC:
20F65 Geometric group theory
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
20E08 Groups acting on trees
53C35 Differential geometry of symmetric spaces
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