# Kissing numbers of closed hyperbolic manifolds

## TL;DR

This paper establishes upper bounds on the number of shortest and primitive closed geodesics in closed hyperbolic manifolds, linking geometric properties like volume and systole, using the Selberg trace formula.

## Contribution

It generalizes Parlier's theorem from surfaces to higher dimensions and provides uniform bounds for geodesics in all closed hyperbolic manifolds with bounded geometry.

## Key findings

- Upper bounds for shortest closed geodesics in terms of volume and systole
- Uniform bounds on primitive geodesics within length intervals
- Application of the Selberg trace formula to hyperbolic geometry

## Abstract

We prove an upper bound for the number of shortest closed geodesics in a closed hyperbolic manifold of any dimension in terms of its volume and systole, generalizing a theorem of Parlier for surfaces. We also obtain bounds on the number of primitive closed geodesics with length in a given interval that are uniform for all closed hyperbolic manifolds with bounded geometry. The proofs rely on the Selberg trace formula.