2001/11/14 by Tsachik Gelander, Gelander, Tsachik
Mathematics · #53C30 #53C35 #Advanced Algebra and Geometry #FOS: Mathematics #Geometry and complex manifolds #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.GR #msc:53C30 #msc:53C35
paper · pdf · doi:10.48550/arxiv.math/0111165
54 pages, amscd
openalex publication_date 2001/11/14 · arxiv created 2004/03/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider locally symmetric manifolds with a fixed universal covering, and construct for each such manifold M a simplicial complex R whose size is proportional to the volume of M. When M is non-compact, R is homotopically equivalent to M, while when M is compact, R is homotopically equivalent to M\N, where N is a finite union of submanifolds of fairly smaller dimensions. This expresses how the volume controls the topological structure of M, and yields concrete bounds for various finiteness statements which previously had no quantitative proofs. For example, it gives an explicit upper bound for the possible number of locally symmetric manifolds of volume bounded by v>0, and it yields an estimate for the size of a minimal presentation for the fundamental group of a manifold in terms of its volume. It also yields a number of new finiteness results.