The asymptotic behaviour of Heegaard genus
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 larg...
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| Format: | Journal article |
| Language: | English |
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2002
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| _version_ | 1826303821024854016 |
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| author | Lackenby, M |
| author_facet | Lackenby, M |
| author_sort | Lackenby, M |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T06:08:31Z |
| format | Journal article |
| id | oxford-uuid:eea91c34-411e-407b-a497-44d0fadda7ab |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:08:31Z |
| publishDate | 2002 |
| record_format | dspace |
| spelling | oxford-uuid:eea91c34-411e-407b-a497-44d0fadda7ab2022-03-27T11:34:27ZThe asymptotic behaviour of Heegaard genusJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:eea91c34-411e-407b-a497-44d0fadda7abEnglishSymplectic Elements at Oxford2002Lackenby, MLet 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. |
| spellingShingle | Lackenby, M The asymptotic behaviour of Heegaard genus |
| title | The asymptotic behaviour of Heegaard genus |
| title_full | The asymptotic behaviour of Heegaard genus |
| title_fullStr | The asymptotic behaviour of Heegaard genus |
| title_full_unstemmed | The asymptotic behaviour of Heegaard genus |
| title_short | The asymptotic behaviour of Heegaard genus |
| title_sort | asymptotic behaviour of heegaard genus |
| work_keys_str_mv | AT lackenbym theasymptoticbehaviourofheegaardgenus AT lackenbym asymptoticbehaviourofheegaardgenus |