Counting non-uniform lattices

Abstract

In [BGLM] and [GLNP] it was conjectured that if H is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in H of covolume at most x is x (γ(H)+o (1)) log x/ log log x where γ(H) is an explicit constant computable from the (absolute) root system of H. In [BLu] we disproved this conjecture. In this paper we prove that for most groups H the conjecture is actually true if we restrict to counting only non-uniform lattices.

On the localization map in the Galois cohomology of algebraic groups

“…This has applications to lattice counting problems (cf. [BL19]). This article gives criteria for ϕ to be surjective.…”

Non left-orderability of lattices in higher rank semi-simple Lie groups

“…In [46], Witte-Morris and Lifschitz proved some cases in Q-rank one, including the case of non-uniform lattices in products of SL(2, R), their proof uses the bounded generation by unipotents property proven by Carter-Keller, [58]. The case where Γ is cocompact, which in some sense is the most common case among lattices as was shown by Belolipetsky and Lubotzky [8], was the main open problem.…”