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

TL;DR

This paper investigates the asymptotic behavior of spectral invariants of locally symmetric spaces, establishing uniform limit theorems and convergence results using a novel adaptation of Benjamini--Schramm convergence for Riemannian manifolds.

Contribution

It introduces a new framework for analyzing the limits of spectral invariants in locally symmetric spaces via BS-convergence and extends existing theorems with uniform versions and quantitative estimates.

Findings

BS-convergence implies convergence of spectral measures and invariants.

Higher rank simple Lie groups have a unique BS-limit, the universal cover.

Quantitative estimates on Betti number convergence for congruence covers.

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 (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 implies convergence, in an appropriate sense, of the associated normalized relative Plancherel measures. 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--St\"{u}ck--Zimmer theorem and Kazhdan's property (T), to obtain a complete understanding of the space of IRSs of $G$.