Uniform Estimates for Closed Geodesics and Homology on Finite Area Hyperbolic Surfaces

In this note, we study the distribution of closed geodesics in homology on a finite area hyperbolic surface. We obtain an estimate which is uniform as the homology class varies, refining an asymptotic formula due to C. Epstein. 0. Introduction Let M be a finite area hyperbolic surface, i.e., the quotient of the hyperbolic plane H by the free action of a group of isometries such that the fundamental domain has finite area. It is well-known that if we define π(T ) to be the number of (prime) closed geodesics on M of length at most T then limT→∞ e−TTπ(T ) = 1. A more delicate problem is to estimate the number of closed geodesics lying in a prescribed homology class. Here there are striking differences depending on whether or not M is compact. The compact case has been studied in [11],[15] and [18]; here we shall concentrate on the case where M has at least one cusp. Suppose that M has genus g and p+1 cusps. Then M has area μ(M) = 2π(2g+ p− 1) and H1(M,Z) ∼= Z. We shall write a typical element of H1(M,Z) as (α, β), where α ∈ Z and β ∈ Z, and use π(α,β)(T ) to denote the number of (prime) closed geodesics in (α, β) of length at most T . Epstein [4] has shown that lim T→∞ T π(α,β)(T ) eT = 1 2g+1 ( 2p p ) (2g + p− 1). (0.1) In this paper, we shall be interested in refining Epstein’s result to obtain a uniform estimate as the class (α, β) is allowed to vary. This is contained in the following theorem. Theorem 1. Let M be a finite area hyperbolic surface of genus g with p+ 1 cusps. Then there exists a strictly positive definite 2g × 2g matrix A of inner products of cusp forms such that lim T→∞ sup (α,β)∈Zp+2g ∣∣∣∣T π(α,β)(T ) eT − (2g + p− 1) 2g−p+1 e−〈β,A−1β〉/4TI ( 2μ(M)α T )∣∣∣∣ = 0, The author was supported by an EPSRC Advanced Research Fellowship. Typeset by AMS-TEX 1