Non-compact arithmetic manifolds have simple homotopy type

Abstract

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 non-compact type without Euclidean de Rham factors, there are constants a=a(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 a 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.