A Generalization of the Prime Geodesic Theorem to Counting Conjugacy Classes of Free Subgroups

Abstract

The classical prime geodesic theorem (PGT) gives an asymptotic formula (as $x$ tends to infinity) for the number of closed geodesics with length at most $x$ on a hyperbolic manifold $M$. Closed geodesics correspond to conjugacy classes of $\pi_1(M)=\Gamma$ where $\Gamma$ is a lattice in $G=SO(n,1)$. The theorem can be rephrased in the following format. Let $X(\Z,\Gamma)$ be the space of representations of $\Z$ into $\Gamma$ modulo conjugation by $\Gamma$. $X(\Z,G)$ is defined similarly. Let $\pi: X(\Z,\Gamma)\to X(\Z,G)$ be the projection map. The PGT provides a volume form $vol$ on $X(\Z,G)$ such that for sequences of subsets $\{B_t\}$, $B_t \subset X(\Z,G)$ satisfying certain explicit hypotheses, $|\pi^{-1}(B_t)|$ is asymptotic to $vol(B_t)$. We prove a statement having a similar format in which $\Z$ is replaced by a free group of finite rank under the additional hypothesis that $n=2$ or 3.