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

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 $\Gamma\backslash G/K$ 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 $G$.