On volumes of quasi-arithmetic hyperbolic lattices

2016/03/31 by Vincent Emery
Mathematics · #Algebra over a field #Arithmetic #Corollary #Geometric and Algebraic Topology #Geometry and complex manifolds #Hyperbolic function #Hyperbolic manifold #Lattice (music) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Pure mathematics #Relatively hyperbolic group #math.KT #math.MG #math.NT

paper · pdf · doi:10.1007/s00029-017-0308-8

published as Sel. Math. New Ser. (2017) 23: 2849 · 11 pages. Final version. To appear in Selecta math

arxiv created 2017/01/31 · openalex publication_date 2017/02/06 · arxiv updated 2018/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We prove that the covolume of any quasi-arithmetic hyperbolic lattice (a notion that generalizes the definition of arithmetic subgroups) is a rational multiple of the covolume of an arithmetic subgroup. As a corollary, we obtain a good description for the shape of the volumes of most of the known hyperbolic n-manifolds with n>3.

Citations

• Volumes of hyperbolic manifolds and mixed Tate motives
• Non-arithmetic groups in Lobachevsky spaces
• Stable real cohomology of arithmetic groups
• The Arithmetic of Hyperbolic 3-Manifolds