Growth of Simple Closed Geodesics in Finite-Volume Hyperbolic Manifolds

Abstract

Let $M$ be a finite-volume hyperbolic $d$-manifold with $d\ge 3$. 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.

21 pages. Explicit polynomials derived from cross-ratio for closed hyperbolic 3-manifolds were replaced by implicit polynomials which work for all dimensions. Counting of simple closed geodesics was replaced by counting of nonsimple ones, which simplifies the paper. Theorem 1.2 (previously 1.3) now works for finite-volume setting as well