Benjamini--Schramm convergence of arithmetic locally symmetric spaces

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.

💡 Research Summary

The paper establishes that for any sequence of pairwise non‑conjugate arithmetic lattices Γₙ in a non‑compact semisimple Lie group G (with symmetric space X = G/K), the volume of the R‑thin part of the locally symmetric spaces Γₙ\X becomes negligible compared to the total volume as n → ∞. This property is precisely the Benjamini–Schramm (BS) convergence of the manifolds to the universal cover X.

The authors treat two fundamentally different regimes of the trace field kₙ = ℚ(Tr Γₙ). If the degrees dₙ =