Unimodularity of invariant random subgroups

Biringer, Ian; Tamuz, Omer

doi:10.1090/tran/6755

Cited by 9 publications

(10 citation statements)

References 32 publications

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“…They implicitly ask if the same holds in the general case. The same question also appeared in the introduction of [BT14] and in [TD12,§7]. We prove the following statement which yields a positive answer as a corollary.…”

Section: Introductionsupporting

confidence: 57%

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Amenable invariant random subgroups

et al. 2016

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Section: Introductionsupporting

confidence: 57%

“…We refer to [AGV12] for historical background on IRSs before the name was coined. Since this first appearance, IRSs appeared in several papers, for example in [ABB + 12] and [BT14] where recent references are given in the introduction.…”

Section: Introductionmentioning

confidence: 99%

Amenable invariant random subgroups

et al. 2016

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“…Graphs and IRSs of discrete groups. All the material here is well-known; for more information we refer the reader to [2,10,25].…”

Section: 4mentioning

confidence: 99%

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

et al. 2017

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Abstract. We study the asymptotic behaviour 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.A basic idea is to adapt the notion of Benjamini-Schramm convergence (BSconvergence), originally introduced for sequences of finite graphs of bounded degree, to sequences of Riemannian manifolds, and analyze the possible limits. We show that BS-convergence of locally symmetric spaces Γ\G/K implies convergence, in an appropriate sense, of the normalized relative Plancherel measures associated to L 2 (Γ\G). This then yields convergence of normalized multiplicities of unitary representations, Betti numbers and other spectral invariants. On the other hand, when the corresponding Lie group G is simple and of real rank at least two, we prove that there is only one possible BS-limit, i.e. when the volume tends to infinity, locally symmetric spaces always BSconverge to their universal cover G/K. This leads to various general uniform results.When restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of BS-convergence which in turn implies 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 Subgroups. For higher rank simple Lie groups G, we exploit rigidity theory, and in particular the Nevo-Stück-Zimmer theorem and Kazhdan's property (T), to obtain a complete understanding of the space of IRSs of G.

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“…Transport de masse. Le contenu de cette section vient de [10] (voir aussi [3]). Si ν est un IRS de PSL 2 (R) sans atome, on note µ ν la mesure poussée en avant sur l'espace S des surfaces hyperboliques pointées.…”

Section: Topologie De La Dimension Trois Rappelons Les Définitions Dunclassified

Géométrie et topologie des variétés hyperboliques de grand volume

Raimbault¹

2014

Séminaire De Théorie Spectrale Et Géométrie

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Cet article est un survol autour de deux prépublications récentes [1] et [39], qui se posent la question de l'étude de certains invariants topologiques et géométriques dans des suites d'espaces localement symétriques dont le volume tend vers l'infini. On donne aussi quelques applicationsà divers modèles de surfaces aléatoires. arXiv:1411.0889v1 [math.GT] 4 Nov 2014 4. Restriction de la topologie de Hausdorff ; cette topologie aété initialement considérée par Claude Chabauty dans [14] ; une description des ouverts est donnée dans [1, Section 2].

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Unimodularity of invariant random subgroups

Cited by 9 publications

References 32 publications

Amenable invariant random subgroups

Amenable invariant random subgroups

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

Géométrie et topologie des variétés hyperboliques de grand volume

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