N ov 2 00 1 Non - compact arithmetic manifolds have simple homotopy type preprint

We formulate a conjecture that arithmetic locally symmetric manifolds have simple homotopy type, and prove it for the non-compact case. More precisely, we show that, for any symmetric space S of noncompact type without Euclidean de Rham factors, there are constants α = α(S) and d = d(S) such that any non-compact arithmetic manifold, locally isometric to S, is homotopically equivalent to a simplicial complex whose vertices degrees are bounded by d, and its number of vertices is bounded by α times the Riemannian volume. It is very likely that such a result holds also for compact arithmetic manifolds. We conclude that, for any fixed universal covering, S, other then the hyperbolic plane, there are at most V CV irreducible non-compact arithmetic manifolds with volume ≤ V , where C = C(S) is a constant depending on S. Since higher rank irreducible locally symmetric manifolds of finite volume are always arithmetic, our result quantifies the number of them which are non-compact.