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Reference: Abert, M, Bergeron, N, Biringer, I et al., (2017). On the growth of $L^2$-invariants for sequences of lattices in Lie groups. Annals of Mathematics, 185 (3), 711-790.Citable link to this page:

 

On the growth of $L^2$-invariants for sequences of lattices in Lie groups

Abstract: We study the asymptotic behaviour of Betti numbers, twisted torsion and otherspectral invariants of sequences of locally symmetric spaces. Our main resultsare uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delormeand various other limit multiplicity theorems. A basic idea is to adapt the notion of Benjamini--Schramm convergence(BS-convergence), originally introduced for sequences of finite graphs ofbounded degree, to sequences of Riemannian manifolds, and analyze the possiblelimits. We show that BS-convergence of locally symmetric spaces impliesconvergence, in an appropriate sense, of the associated normalized relativePlancherel measures. This then yields convergence of normalized multiplicitiesof unitary representations, Betti numbers and other spectral invariants. On theother hand, when the corresponding Lie group $G$ is simple and of real rank atleast two, we prove that there is only one possible BS-limit, i.e. when thevolume tends to infinity, locally symmetric spaces always BS-converge to theiruniversal cover $G/K$. This leads to various general uniform results. When restricting to arbitrary sequences of congruence covers of a fixedarithmetic manifold we prove a strong quantitative version of BS-convergencewhich in turn implies upper estimates on the rate of convergence of normalizedBetti numbers in the spirit of Sarnak--Xue. An important role in our approach is played by the notion of Invariant RandomSubgroups. For higher rank simple Lie groups $G$, we exploit rigidity theory,and in particular the Nevo--St\{u}ck--Zimmer theorem and Kazhdan's property(T), to obtain a complete understanding of the space of IRSs of $G$.

Publication status:PublishedPeer Review status:Peer reviewedVersion:Accepted manuscriptDate of acceptance:2016-12-18 Funder: MTA Renyi “Lendulet” Groups and Graphs Research Group   Funder: National Science Foundation   Funder: Institut Universitaire de France   Funder: European Research Council   Funder: Engineering and Physical Sciences Research Council   Funder: Israel Science Foundation   Notes:© Department of Mathematics, Princeton University. This is the accepted manuscript version of the article. The final version is available online from the Department of Mathematics, Princeton University at: [10.4007/annals.2017.185.3.1]

Bibliographic Details

Publisher: Department of Mathematics, Princeton University

Publisher Website: http://www.math.princeton.edu/

Journal: Annals of Mathematicssee more from them

Publication Website: http://annals.math.princeton.edu/

Volume: 185

Issue: 3

Extent: 711-790

Issue Date: 2017-04-12

pages:711-790Identifiers

Uuid: uuid:79ce917f-5c41-40b1-beea-824588db0268

Urn: uri:79ce917f-5c41-40b1-beea-824588db0268

Pubs-id: pubs:354423

Issn: 0003-486X

Eissn: 1939-8980

Doi: https://doi.org/10.4007/annals.2017.185.3.1 Item Description

Type: journal-article;

Version: Accepted manuscriptKeywords: math.RT math.RT math.DG math.GR math.KT Article

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Author: Abert, M - - - Bergeron, N - - - Biringer, I - - - Gelander, T - - - Nikolov, N - Oxford, MPLS, Mathematical Institute University

Source: https://ora.ox.ac.uk/objects/uuid:79ce917f-5c41-40b1-beea-824588db0268



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