A finiteness theorem for hyperbolic 3-manifolds

Abstract. We prove that there are only finitely many closed hy-perbolic 3-manifolds with injectivity radius and first eigenvalue of the Laplacian bounded below whose fundamental groups can be generated by a given number of elements. 1 A common pursuit in differential geometry is to bound the number of homotopy types of closed n-manifolds that admit a Riemannian metric with controlled geometry: for instance, one often specifies constraints on diameter, curvature, volume or injectivity radius. If one considers only locally symmetric manifolds, these finiteness theorems combine with Mostow’s rigidity theorem to yield much stronger results. For example, Wang’s finiteness theorem [18] asserts that for every d ≥ 4 and V positive, there are only finitely many isometry classes of hyperbolic d-manifolds M with volume vol(M) ≤ V. In dimension 3, Wang’s finiteness theorem still holds if, in addition to the volume bound, one only considers manifolds M with injectivity radius inj(M) ≥ > 0. Our goal is to provide a finiteness result for hyperbolic 3-manifolds in terms of injectivity radius, first eigenvalue of the Laplacian and rank of the fundamental group. Here, the rank of a group is the minimal number of elements needed to generate it. We prove: Theorem 1.1. For every , δ, k> 0, there are only finitely many isome-try classes of hyperbolic 3-manifolds M with injectivity radius inj(M) ≥ δ, first eigenvalue of the Laplacian λ1(M) ≥ δ and rank(pi1(M)) ≤ k. In the final section of this note we sketch the proofs of stronger finiteness results under the assumption that the involved manifolds are arithmetic. The idea of the proof of Theorem 1.1 is as follows. Assume that we have a sequence of pairwise distinct hyperbolic 3-manifolds (Mi