Growth of Simple Closed Geodesics in Finite-Volume Hyperbolic Manifolds

Abstract

Let $M$ be a finite-volume hyperbolic $d$-manifold with $d\ge3$. We prove that the number of primitive nonsimple closed geodesics has exponential growth rate strictly smaller than $d-1$. Consequently, asymptotically almost every primitive closed geodesic in $M$ is simple. In contrast, we show that on an arithmetic hyperbolic manifold of type~I, the unit vectors tangent to nonsimple closed geodesics are dense in the unit tangent bundle.