On the growth of $L^2$-invariants for sequences of lattices in Lie groups
We study the asymptotic behavior of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems. The idea is to a...
| Main Authors: | , , , , , , |
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| Format: | Journal article |
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2012
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| 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 study the asymptotic behavior of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems. The idea is to adapt Benjamini-Schramm convergence (BS-convergence), originally introduced for sequences of finite graphs of bounded degree, to sequences of Riemannian manifolds. Exploiting the rigidity theory of higher rank lattices, we show that when volume tends to infinity, higher rank locally symmetric spaces BS-converge to their universal cover. We prove that BS-convergence implies a convergence of certain spectral invariants, the Plancherel measures. This leads to convergence of volume normalized multiplicities of unitary representations and Betti numbers. We also prove a strong quantitative version of BS-convergence for arbitrary sequences of congruence covers of a fixed arithmetic manifold. This leads to upper estimates on the rate of convergence of normalized Betti numbers in the spirit of Sarnak-Xue. An important role in our approach is played by the notion of Invariant Random Subroups. For higher rank simple Lie groups $G$, exploiting rigidity theory, and in particular the Nevo-Stuck-Zimmer theorem and Kazhdan`s property (T), we are able to analyze the space of IRSs of $G$. In rank one, the space of IRSs is much richer. We build some explicit 2 and 3-dimensional real hyperbolic IRSs that are not induced from lattices and employ techniques of Gromov--Piatetski-Shapiro to construct similar examples in higher dimension. |
| first_indexed | 2024-03-07T02:34:44Z |
| format | Journal article |
| id | oxford-uuid:a869fd45-0dce-463c-81a3-9aee19738d14 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T02:34:44Z |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oxford-uuid:a869fd45-0dce-463c-81a3-9aee19738d142022-03-27T03:01:23ZOn the growth of $L^2$-invariants for sequences of lattices in Lie groupsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:a869fd45-0dce-463c-81a3-9aee19738d14Symplectic Elements at Oxford2012Abert, MBergeron, NBiringer, IGelander, TNikolov, NRaimbault, JSamet, IWe study the asymptotic behavior of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems. The idea is to adapt Benjamini-Schramm convergence (BS-convergence), originally introduced for sequences of finite graphs of bounded degree, to sequences of Riemannian manifolds. Exploiting the rigidity theory of higher rank lattices, we show that when volume tends to infinity, higher rank locally symmetric spaces BS-converge to their universal cover. We prove that BS-convergence implies a convergence of certain spectral invariants, the Plancherel measures. This leads to convergence of volume normalized multiplicities of unitary representations and Betti numbers. We also prove a strong quantitative version of BS-convergence for arbitrary sequences of congruence covers of a fixed arithmetic manifold. This leads to upper estimates on the rate of convergence of normalized Betti numbers in the spirit of Sarnak-Xue. An important role in our approach is played by the notion of Invariant Random Subroups. For higher rank simple Lie groups $G$, exploiting rigidity theory, and in particular the Nevo-Stuck-Zimmer theorem and Kazhdan`s property (T), we are able to analyze the space of IRSs of $G$. In rank one, the space of IRSs is much richer. We build some explicit 2 and 3-dimensional real hyperbolic IRSs that are not induced from lattices and employ techniques of Gromov--Piatetski-Shapiro to construct similar examples in higher dimension. |
| spellingShingle | Abert, M Bergeron, N Biringer, I Gelander, T Nikolov, N Raimbault, J Samet, I On the growth of $L^2$-invariants for sequences of lattices in Lie groups |
| title | On the growth of $L^2$-invariants for sequences of lattices in Lie
groups |
| title_full | On the growth of $L^2$-invariants for sequences of lattices in Lie
groups |
| title_fullStr | On the growth of $L^2$-invariants for sequences of lattices in Lie
groups |
| title_full_unstemmed | On the growth of $L^2$-invariants for sequences of lattices in Lie
groups |
| title_short | On the growth of $L^2$-invariants for sequences of lattices in Lie
groups |
| title_sort | on the growth of l 2 invariants for sequences of lattices in lie groups |
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