The asymptotic behaviour of Heegaard genus

Journal article

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.

Authors: Lackenby, M

• Publication status: Published
• Journal: MATHEMATICAL RESEARCH LETTERS
• Volume: 11
• Issue: 2-3
• Pages: 139-149
• Publication date: 2002-10-21
• DOI: 10.4310/MRL.2004.v11.n2.a1
• EISSN: 1945-001X
• ISSN: 1073-2780
• Language: English
• Keywords: 57N10; 57M10; math.GT