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 $X(\mathbb{Z},\Gamma )$ be the space of representations of $\mathbb{Z}$ into $\Gamma$ modulo conjugation by $\Gamma$. $X(\mathbb{Z},G)$ is defined similarly. Let $\pi :X(\mathbb{Z},\Gamma )\to X(\mathbb{Z},G)$ be the projection map. The PGT provides a volume form $vol$ on $X(\mathbb{Z},G)$ such that for sequences of subsets $\{B_t\}$, $B_t\subset X(\mathbb{Z},G)$ satisfying certain explicit hypotheses, $|{\pi }^{-1}(B_t)|$ is asymptotic to $vol(B_t)$. We prove a statement having a similar format in which $\mathbb{Z}$ is replaced by a free group of finite rank under the additional hypothesis that $n=2$ or 3.

32 pages, 5 figures. This is the second version. The introduction has been expanded and two new examples inserted