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 π 1 (M ) = Γ where Γ is a lattice in G = SO(n, 1). The theorem can be rephrased in the following format. Let X(Z, Γ) be the space of representations of Z into Γ modulo conjugation by Γ. X(Z, G) is defined similarly. Let π : X(Z, Γ) → 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 ⊂ X(Z, G) satisfying certain explicit hypotheses, |π −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.MSC: 20E09, 20F69, 37E35, 51M10