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

TL;DR

This paper studies how orbits of geometrically finite hyperbolic groups distribute on the boundary, showing they are equidistributed with respect to Patterson-Sullivan measure under certain conditions, using flows on the tangent bundle.

Contribution

It establishes equidistribution of orbits for geometrically finite hyperbolic groups via solvable flow analysis, extending previous results with a new approach.

Findings

Orbits are equidistributed with Patterson-Sullivan measure.

The approach uses solvable flow equidistribution on tangent bundles.

Results apply to geometrically finite hyperbolic groups.

Abstract

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.