Closed geodesics and holonomies for Kleinian manifolds

TL;DR

This paper proves the equidistribution of holonomy classes around closed geodesics in certain hyperbolic manifolds, providing quantitative results and interpreting eigenvalue distributions in complex hyperbolic geometry.

Contribution

It establishes new quantitative equidistribution results for holonomy classes in geometrically finite hyperbolic manifolds, extending previous lattice cases.

Findings

Holonomy classes are equidistributed around closed geodesics.

Quantitative bounds are provided for hyperbolic manifolds with large critical exponents.

Eigenvalues of Kleinian groups are shown to distribute uniformly in the complex plane.

Abstract

For a rank one Lie group G and a Zariski dense and geometrically finite subgroup $\Gamma$ of G, we establish equidistribution of holonomy classes about closed geodesics for the associated locally symmetric space. Our result is given in a quantitative form for real hyperbolic geometrically finite manifolds whose critical exponents are big enough. In the case when G=PSL(2, C), our results can be interpreted as the equidistribution of eigenvalues of $\Gamma$ in the complex plane. When $\Gamma$ is a lattice, this result was proved by Sarnak and Wakayama in 1999.