Benjamini--Schramm convergence of arithmetic locally symmetric spaces

TL;DR

This paper demonstrates that the thin regions of certain arithmetic locally symmetric spaces occupy negligible volume, leading to asymptotic insights into their Betti numbers.

Contribution

It establishes the Benjamini--Schramm convergence for arithmetic locally symmetric spaces and analyzes their geometric and topological properties.

Findings

Thin parts have negligible volume in the limit

Betti numbers exhibit specific asymptotic behavior

Supports conjectures on geometric limits of these spaces

Abstract

We prove that the thin parts of arithmetically defined locally symmetric space take up a negligible part of their volume and deduce asymptotic results on their Betti numbers.