Equidistribution of hyperbolic groups in homogeneous spaces

TL;DR

This paper proves that infinite orbits of Zariski dense hyperbolic groups become evenly spread out in homogeneous spaces, with measures along spheres converging to the Haar measure.

Contribution

It establishes the equidistribution of hyperbolic group orbits in homogeneous spaces, extending understanding of their geometric and dynamical properties.

Findings

Infinite orbits of Zariski dense hyperbolic groups equidistribute in homogeneous spaces.

Measures averaged along spheres in the Cayley graph converge to Haar measure.

Provides new insights into the dynamics of hyperbolic groups.

Abstract

We prove that infinite orbits of Zariski dense hyperbolic groups equidistribute in homogeneous spaces, in the sense that the family of measures obtained by averaging along spheres in the Cayley graph converges to Haar measure.