2001/11/26 by Tsachik Gelander, Gelander, Tsachik
Mathematics · #53C30 #53C35 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.DG #math.GR #msc:53C30 #msc:53C35
paper · pdf · doi:10.48550/arxiv.math/0111261
17 pages, amscd
arxiv created 2001/11/26 · openalex publication_date 2001/11/26 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.