Chebotarev-type Theorems in Homology Classes

We describe how closed geodesics lying in a prescribed homology class on a negatively curved manifold split when lifted to a finite cover. This generalizes a result of Zelditch in the case of compact hyperbolic surfaces. 0. Introduction Given a compact manifold of negative curvature, there are geometric analogues of the Chebotarev Theorem in algebraic number theory due to Sunada [13] (cf. also Parry and Pollicott [8] for the generalization to Axiom A flows). More precisely, given a finite Galois cover of the manifold, these theorems describe the proportion of closed geodesics which lift in a prescribed way to the cover. In this geometric setting, it is also natural to consider infinite covers and, in particular, the number of closed geodesics lying in a prescribed homology class has been studied by Katsuda and Sunada [4], Phillips and Sarnak [9], Katsuda [3], Lalley [7] and Pollicott [10] (with generalizations to Anosov flows by Katsuda and Sunada [5] and Sharp [12]). In this note we shall combine these points of view, generalizing a result of Zelditch for hyperbolic Riemann surfaces [14]. Let M be a compact smooth Riemannian manifold with negative curvature. Let M̃ be a finite Galois covering of M with covering group G. For a closed geodesic γ on M , let l(γ) denote its length, 〈γ〉 its Frobenius class in G and [γ] its homology class in H = H1(M,Z). We shall examine how the closed geodesics lying in a fixed homology class α ∈ H, split when lifted to M̃ . More precisely, for a conjugacy class C in G, we study the asymptotics of π(T, α, C) = Card{γ : l(γ) ≤ T, [γ] = α, 〈γ〉 = C}. The problem is complicated by the fact that that, in general, [γ] and 〈γ〉 are not independent quantities. This occurs if the abelian quotient group G/[G,G] is non-trivial, since this group is also a quotient of H, the maximal abelian covering group of M . Let πG : G → G/[G,G] and πH : H → G/[G,G] be the natural projections. In particular, the image πG(C) of a conjugacy class C ⊂ G is a single element in G/[G,G] and if πG(C) 6= πH(α) then π(T, α, C) = 0, for all values of T . On the other hand, we have the following result, which extends work of Zelditch for Riemann surfaces [14]. Typeset by AMS-TEX 1 2 MARK POLLICOTT AND RICHARD SHARP Theorem 1. If πG(C) 6= πH(α) then π(T, α, C) is identically zero. If πG(C) = πH(α) then π(T, α, C) π(T, α) → ∣∣∣∣ G [G,G] ∣∣∣∣ |C| |G| as T → +∞, where π(T, α) = Card{γ : l(γ) ≤ T, [γ] = α}. Example. Let G be a finite nonabelian nilpotent group, then [G,G] 6= G. For definiteness, we can let G = {±1,±i,±j,±k} be the Quarternion group of eight elements, then [G,G] = ±1. Let Γ = 〈a1, a2, b1, b2 : a1b1a 1 b 1 a2b2a 2 b 2 = 1〉 be a cocompact Fuchsian group. Define a homomorphism φ : Γ → G by setting φ(a1) = i, φ(a2) = j and φ(b1) = φ(b2) = 1 and extending this to Γ. We can then define a normal subgroup by Γ0 = ker(φ). If we set M = H /Γ0 and M̃ = H /Γ0 then M̃ is a finite cover of M with covering group G. Let us consider a closely related problem. Consider the frame flow ft : FM → FM on the space of orthonormal frames above M . This is a SO(n− 1)-extension for the geodesic flow. Changing notation slightly, let γ be a periodic orbit of the geodesic flow, to which we associate a holonomy Θ(γ) ∈ SO(n − 1) which comes from a reference frame being transported around γ. This is defined up to conjugacy. In [8] it was shown that the holonomies were equidistributed on SO(n − 1). The following shows that the corresponding result holds for geodesics in a fixed homology class. (Recall that a class function is a function which is constant on conjugacy classes.) Theorem 2. Let F : SO(n− 1) → R be a class function. Then 1 π(T, α) ∑ l(γ)≤T [γ]=α F (Θ(γ)) → ∫ Fdλ, as T → +∞, where λ denotes the Haar measure on SO(n− 1). 1. Preliminaries Let M be a compact smooth manifold equipped with a Riemannian metric of negative curvature and let X denote its universal cover. (In the special case where M is a surface with constant curvature −1, X is the hyperbolic plane H.) Then there is a discrete group of isometries Γ ∼= π1(M) of X such that M = X/Γ. Now let Γ0 be a normal subgroup of Γ with finite index. Then M̃ = X/Γ0 is a finite (Galois) covering of M , with covering group G = Γ/Γ0 (i.e., G acts transitively on the fibres above each point in M). There is a natural dynamical system, the geodesic flow, associated to M . Let SM denote the unit-tangent bundle of M and, for (x, v) ∈ SM , let γ : R → M be the unique unit-speed geodesic with γ(0) = x and γ̇(0) = v. Then the geodesic flow φt : SM → SM is defined by φt(x, v) = (γ(t), γ̇(t)) and we shall write h for its topological entropy. There is a one-to-one correspondence between periodic φ-orbits and directed closed geodesics on M . The fact that M is negatively curved ensures that the geodesic flow is an Anosov flow and that h > 0. This will enable us to use results proved in the context of Anosov flows in this setting. CHEBOTAREV-TYPE THEOREMS IN HOMOLOGY CLASSES 3 We shall make use of L-functions defined with respect to certain representations of Γ = π1(M). Let ρ : Γ → U(d) be a unitary representation of Γ. We define an L-function L(s, ρ) by the product formula