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


Abert, M
Bergeron, N
Biringer, I
Gelander, T
Nikolov, N
Raimbault, J
Samet, I

Publication Date: 

12 April 2017


Annals of Mathematics

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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.

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Publication Type: 

Journal Article