Large groups, Property (tau) and the homology growth of subgroups
We investigate the homology of finite index subgroups G_i of a given finitely presented group G. Specifically, we examine d_p(G_i), which is the dimension of the first homology of G_i, with mod p coefficients. We say that a collection of finite index subgroups {G_i} has linear growth of mod p homolo...
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| Format: | Journal article |
| Language: | English |
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2005
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| _version_ | 1826266650371948544 |
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| author | Lackenby, M |
| author_facet | Lackenby, M |
| author_sort | Lackenby, M |
| collection | OXFORD |
| description | We investigate the homology of finite index subgroups G_i of a given finitely presented group G. Specifically, we examine d_p(G_i), which is the dimension of the first homology of G_i, with mod p coefficients. We say that a collection of finite index subgroups {G_i} has linear growth of mod p homology if the infimum of d_p(G_i)/[G:G_i] is positive. We show that if this holds and each G_i is normal in its predecessor and has index a power of p, then one of the following possibilities must be true: G is large (that is, some finite index subgroup admits a surjective homomorphism onto a non-abelian free group) or G has Property (tau) with respect to {G_i}. The arguments are based on the geometry and topology of finite 2-complexes. This has several consequences. It implies that if the pro-p completion of a finitely presented group G has exponential subgroup growth, then G has Property (tau) with respect to some nested sequence of finite index subgroups. It also has applications to low-dimensional topology. We use it to prove that a group-theoretic conjecture of Lubotzky-Zelmanov would imply the following: any lattice in PSL(2,C) with torsion is large. We also relate linear growth of mod p homology to the existence of certain important error-correcting codes: those that are `asymptotically good', which means that they have large rate and large Hamming distance. |
| first_indexed | 2024-03-06T20:42:10Z |
| format | Journal article |
| id | oxford-uuid:34a45276-c60a-42f8-9ac5-399a185d17a4 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T20:42:10Z |
| publishDate | 2005 |
| record_format | dspace |
| spelling | oxford-uuid:34a45276-c60a-42f8-9ac5-399a185d17a42022-03-26T13:27:15ZLarge groups, Property (tau) and the homology growth of subgroupsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:34a45276-c60a-42f8-9ac5-399a185d17a4EnglishSymplectic Elements at Oxford2005Lackenby, MWe investigate the homology of finite index subgroups G_i of a given finitely presented group G. Specifically, we examine d_p(G_i), which is the dimension of the first homology of G_i, with mod p coefficients. We say that a collection of finite index subgroups {G_i} has linear growth of mod p homology if the infimum of d_p(G_i)/[G:G_i] is positive. We show that if this holds and each G_i is normal in its predecessor and has index a power of p, then one of the following possibilities must be true: G is large (that is, some finite index subgroup admits a surjective homomorphism onto a non-abelian free group) or G has Property (tau) with respect to {G_i}. The arguments are based on the geometry and topology of finite 2-complexes. This has several consequences. It implies that if the pro-p completion of a finitely presented group G has exponential subgroup growth, then G has Property (tau) with respect to some nested sequence of finite index subgroups. It also has applications to low-dimensional topology. We use it to prove that a group-theoretic conjecture of Lubotzky-Zelmanov would imply the following: any lattice in PSL(2,C) with torsion is large. We also relate linear growth of mod p homology to the existence of certain important error-correcting codes: those that are `asymptotically good', which means that they have large rate and large Hamming distance. |
| spellingShingle | Lackenby, M Large groups, Property (tau) and the homology growth of subgroups |
| title | Large groups, Property (tau) and the homology growth of subgroups |
| title_full | Large groups, Property (tau) and the homology growth of subgroups |
| title_fullStr | Large groups, Property (tau) and the homology growth of subgroups |
| title_full_unstemmed | Large groups, Property (tau) and the homology growth of subgroups |
| title_short | Large groups, Property (tau) and the homology growth of subgroups |
| title_sort | large groups property tau and the homology growth of subgroups |
| work_keys_str_mv | AT lackenbym largegroupspropertytauandthehomologygrowthofsubgroups |