Counting lattices in simple Lie groups: the positive characteristic case

In this article we prove a conjecture of A. Lubotzky: let G = G0(K), where K is a local field of characteristic p ≥ 5, G0 is a simply connected, absolutely almost simple K-group of K-rank at least 2. We give the rate of growth of ρx(G) := |{Γ ⊆ G|Γ a lattice in G, vol(G/Γ) ≤ x}/ ∼ |, where Γ1 ∼ Γ2 if and only if there is an abstract automorphism θ of G such that Γ2 = θ(Γ1). We also study the rate of subgroup growth sx(Γ) of any lattice Γ in G. As a result we show that these two functions have the same rate of growth which proves Lubotzky’s conjecture. Along the way, we also study the rate of growth of the number of equivalence classes of maximal lattices in G with covolume at most x.