Counting common perpendicular arcs in negative curvature

TL;DR

This paper derives an asymptotic formula for counting common perpendiculars between convex subsets in negatively curved manifolds, with applications to Kleinian groups, using mixing properties of geodesic flows.

Contribution

It introduces a new asymptotic counting method for common perpendiculars in negatively curved manifolds and applies it to Kleinian groups, extending previous results with error estimates.

Findings

Asymptotic formula for common perpendiculars as length tends to infinity

Equidistribution of tangent vectors at endpoints of perpendiculars

Application to counting components of Kleinian group domains

Abstract

Let $D^-$ and $D^+$ be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as $t\to+\infty$ for the number of common perpendiculars of length at most $t$ from $D^-$ to $D^+$, counted with multiplicities, and we prove the equidistribution in the outer and inner unit normal bundles of $D^-$ and $D^+$ of the tangent vectors at the endpoints of the common perpendiculars. When the manifold is compact with exponential decay of correlations or arithmetic with finite volume, we give an error term for the asymptotic. As an application, we give an asymptotic formula for the number of connected components of the domain of discontinuity of Kleinian groups as their diameter goes to $0$.