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May 2017 On the growth of $L^2$-invariants for sequences of lattices in Lie groups
Miklos Abert, Nicolas Bergeron, Ian Biringer, Tsachik Gelander, Nikolay Nikolav, Jean Raimbault, Iddo Samet
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Ann. of Math. (2) 185(3): 711-790 (May 2017). DOI: 10.4007/annals.2017.185.3.1


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 (BS-convergence), 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 $\Gamma\backslash G/K$ implies convergence, in an appropriate sense, of the normalized relative Plancherel measures associated to $L^2 (\Gamma\backslash G)$.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 BS-converge 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 IRS's of $G$.


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Miklos Abert. Nicolas Bergeron. Ian Biringer. Tsachik Gelander. Nikolay Nikolav. Jean Raimbault. Iddo Samet. "On the growth of $L^2$-invariants for sequences of lattices in Lie groups." Ann. of Math. (2) 185 (3) 711 - 790, May 2017.


Published: May 2017
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2017.185.3.1

Rights: Copyright © 2017 Department of Mathematics, Princeton University


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Vol.185 • No. 3 • May 2017
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