2021/10/28 by Rosemary K. Guzman, Peter B. Shalen, Guzman, Rosemary K. +1
Mathematics · #57K32 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2110.14847
openalex publication_date 2021/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that for every finite-volume hyperbolic 3-manifold M and every prime p we have dim H1(M;Fp)< 168.602\cdotvol M. There are slightly stronger estimates if p = 2 or if M is non-compact. This improves on a result proved by Agol, Leininger and Margalit, which gave the same inequality with a coefficient of 334.08 in place of 168.602. It also improves on the analogous result with a coefficient of about 260, which could have been obtained by combining the arguments due to Agol, Leininger and Margalit with a result due to Böröczky. Our inequality involving homology rank is deduced from a result about the rank of the fundamental group: if M is a finite-volume orientable hyperbolic 3-manifold such that π1(M) is 2-semifree, then rank π1(M)<1+λ0\cdotvol M, where λ0 is a certain constant less than 167.79