Counting commensurability classes of hyperbolic manifolds

2014/01/30 by Tsachik Gelander, Gelander, Tsachik, Arie Levit +1 · 2 citations
Mathematics · #20E07 #20G20 #22E40 #53C30 #53C35 #57N16 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1401.8003

openalex publication_date 2014/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about v^v such manifolds of volume at most v, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, almost all hyperbolic manifolds are non-arithmetic in an appropriate sense. Our method involves a geometric graph-of-spaces construction that relies on arithmetic properties of certain quadratic forms.

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• Counting hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants
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