Abért, Miklós and Bergeron, N. and Biringer, I. and Gelander, T. and Nikolov, N. (2017) On the growth of L2invariants for sequences of lattices in Lie groups. ANNALS OF MATHEMATICS, 185 (3). pp. 711790. ISSN 0003486X

Text
1210.2961v4.pdf Download (751kB)  Preview 
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 DeGeorgeWallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems. A basic idea is to adapt the notion of BenjaminiSchramm 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 BSconvergence of locally symmetric spaces Γ\G/K implies convergence, in an appropriate sense, of the normalized relative Plancherel measures associated to L2(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 BSlimit; 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 BSconvergence, which in turn implies upper estimates on the rate of convergence of normalized Betti numbers in the spirit of SarnakXue. 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 NevoStückZimmer theorem and Kazhdan's property (T), to obtain a complete understanding of the space of IRS's of G. © 2017 Department of Mathematics, Princeton University.
Item Type:  Article 

Subjects:  Q Science / természettudomány > QA Mathematics / matematika 
SWORD Depositor:  MTMT SWORD 
Depositing User:  MTMT SWORD 
Date Deposited:  03 Oct 2017 13:59 
Last Modified:  03 Oct 2017 13:59 
URI:  http://real.mtak.hu/id/eprint/64994 
Actions (login required)
Edit Item 