On the growth of Betti numbers of locally symmetric spaces
We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the L\"uck Approximation Theorem \cite{luck}, which is much stronger than the linear upper bounds o...
| Main Authors: | , , , , , , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
2011
|
| _version_ | 1826266992479305728 |
|---|---|
| author | Abert, M Bergeron, N Biringer, I Gelander, T Nikolov, N Raimbault, J Samet, I |
| author_facet | Abert, M Bergeron, N Biringer, I Gelander, T Nikolov, N Raimbault, J Samet, I |
| author_sort | Abert, M |
| collection | OXFORD |
| description | We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the L\"uck Approximation Theorem \cite{luck}, which is much stronger than the linear upper bounds on Betti numbers given by Gromov in \cite{BGS}. The basic idea is to adapt the theory of local convergence, originally introduced for sequences of graphs of bounded degree by Benjamimi and Schramm, to sequences of Riemannian manifolds. Using rigidity theory we are able to show that when the volume tends to infinity, the manifolds locally converge to the universal cover in a sufficiently strong manner that allows us to derive the convergence of the normalized Betti numbers. |
| first_indexed | 2024-03-06T20:47:21Z |
| format | Journal article |
| id | oxford-uuid:365b113f-2937-4378-a729-3b0529643bcc |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T20:47:21Z |
| publishDate | 2011 |
| record_format | dspace |
| spelling | oxford-uuid:365b113f-2937-4378-a729-3b0529643bcc2022-03-26T13:37:23ZOn the growth of Betti numbers of locally symmetric spacesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:365b113f-2937-4378-a729-3b0529643bccEnglishSymplectic Elements at Oxford2011Abert, MBergeron, NBiringer, IGelander, TNikolov, NRaimbault, JSamet, IWe announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the L\"uck Approximation Theorem \cite{luck}, which is much stronger than the linear upper bounds on Betti numbers given by Gromov in \cite{BGS}. The basic idea is to adapt the theory of local convergence, originally introduced for sequences of graphs of bounded degree by Benjamimi and Schramm, to sequences of Riemannian manifolds. Using rigidity theory we are able to show that when the volume tends to infinity, the manifolds locally converge to the universal cover in a sufficiently strong manner that allows us to derive the convergence of the normalized Betti numbers. |
| spellingShingle | Abert, M Bergeron, N Biringer, I Gelander, T Nikolov, N Raimbault, J Samet, I On the growth of Betti numbers of locally symmetric spaces |
| title | On the growth of Betti numbers of locally symmetric spaces |
| title_full | On the growth of Betti numbers of locally symmetric spaces |
| title_fullStr | On the growth of Betti numbers of locally symmetric spaces |
| title_full_unstemmed | On the growth of Betti numbers of locally symmetric spaces |
| title_short | On the growth of Betti numbers of locally symmetric spaces |
| title_sort | on the growth of betti numbers of locally symmetric spaces |
| work_keys_str_mv | AT abertm onthegrowthofbettinumbersoflocallysymmetricspaces AT bergeronn onthegrowthofbettinumbersoflocallysymmetricspaces AT biringeri onthegrowthofbettinumbersoflocallysymmetricspaces AT gelandert onthegrowthofbettinumbersoflocallysymmetricspaces AT nikolovn onthegrowthofbettinumbersoflocallysymmetricspaces AT raimbaultj onthegrowthofbettinumbersoflocallysymmetricspaces AT sameti onthegrowthofbettinumbersoflocallysymmetricspaces |