Uniform spectral gap and orthogeodesic counting for strong convergence of Kleinian groups

TL;DR

This paper establishes uniform spectral gap results and asymptotic orthogeodesic counting formulas for sequences of geometrically finite hyperbolic manifolds and Kleinian groups, leveraging strong convergence and mixing properties.

Contribution

It introduces a novel approach linking spectral gaps with orthogeodesic counting in strongly convergent Kleinian groups, extending previous results to new geometric settings.

Findings

Convergence of small eigenvalues under strong limits.

Asymptotically uniform counting formulas for orthogeodesics.

Application to Margulis tubes and closed geodesics.

Abstract

We show convergence of small eigenvalues for geometrically finite hyperbolic $n$-manifolds under strong limits. For a class of convergent convex sets in a strongly convergent sequence of Kleinian groups, we use the spectral gap of the limit manifold and the exponentially mixing property of the geodesic flow along the strongly convergent sequence to find asymptotically uniform counting formulas for the number of orthogeodesics between the convex sets. In particular, this provides asymptotically uniform counting formulas (with respect to length) for orthogeodesics between converging Margulis tubes, geodesic loops based at converging basepoints, and primitive closed geodesics.