The asymptotic behaviour of Heegaard genus

2002/10/21 by Marc Lackenby, Lackenby, Marc
Mathematics · #57M10 #57N10 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #math.GT #msc:57M10 #msc:57N10

paper · pdf · doi:10.48550/arxiv.math/0210316

14 pages. Final version, including a new expository section. To appear in Mathematical Research Letters

openalex publication_date 2002/10/21 · arxiv created 2003/11/26 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M be a closed orientable 3-manifold with a negatively curved Riemannian metric. Let M_i be a collection of finite regular covers with degree d_i. (1) If the Heegaard genus of M_i grows more slowly than the square root of d_i, then M_i has positive first Betti number for all sufficiently large i. (2) The strong Heegaard genus of M_i cannot grow more slowly than the square root of d_i. (3) If the Heegaard genus of M_i grows more slowly than the fourth root of d_i, then M_i fibres over the circle for all sufficiently large i. These results provide supporting evidence for the Heegaard gradient conjecture and the strong Heegaard gradient conjecture. As a corollary to (3), we give a necessary and sufficient condition for M to be virtually fibred in terms of the Heegaard genus of its finite covers.

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