Benjamini--Schramm convergence of arithmetic locally symmetric spaces

TL;DR

The paper proves that sequences of pairwise non-conjugate congruence arithmetic lattices Γ_n yield Benjamini–Schramm convergence of the associated locally symmetric spaces Γ_n\X to the universal cover X, showing that thin parts occupy vanishing volume and enabling Betti-number asymptotics via $L^2$-Betti numbers. The authors develop a two-pronged strategy: unbounded trace-field degree cases are handled by quantitative bounds from previous work, while bounded-degree cases are tackled with an adelic trace formula approach, extensive orbital-integral estimates, and Galois-cohomology inputs. They provide detailed local and global estimates for orbital integrals, connect height-counts of semisimple elements to rational conjugacy classes, and derive limit multiplicities that translate into Betti-number asymptotics through Matsushima’s formula. The results unify BS-convergence with spectral and cohomological consequences for arithmetic locally symmetric spaces and address extensions to non-congruence settings, outline optimal-thin-part bounds, and pose open questions about non-congruence BS-convergence and sharper volume bounds for the thin part.

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.

Theorems & Definitions (88)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Conjecture 1.1
• Proposition 2.1
• proof
• Proposition 2.2
• proof