On the distribution of orbits of geometrically finite hyperbolic groups on the boundary (with appendix by Francois Maucourant)

Details On the distribution of orbits of geometrically finite hyperbolic groups on the boundary (with appendix by Francois Maucourant) On the distribution of orbits of geometrically finite hyperbolic groups on the boundary (with appendix by Francois Maucourant)

2010-04-15

arxiv.org/abs/1004.2554v3

Mathematics

Dynamical Systems

14 pages, 1 figure. More references are added. To appear at ETDS

Scientific paper

We investigate the distribution of orbits of a non-elementary discrete hyperbolic group acting on the n-dimensional hyperbolic space and its geometric boundary. In particular, we show that if the group $\Gamma$ admits a finite Bowen-Margulis-Sullivan measure (for instance, if it is geometrically finite), then every $\Gamma$-orbit in the hyperbolic space is equidistributed with respect to the Patterson-Sullivan measure supported on the limit set of $\Gamma$.The appendix by Maucourant is the extension of a part of his thesis where he obtains the same result as a simple application of Roblin's theorem. Our approach is via establishing the equidistribution of solvable flows on the unit tangent bundle of the quotient manifold, which is of independent interest.

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