Counting lattices in simple Lie groups: the positive characteristic case

TL;DR

This paper proves Lubotzky's conjecture by establishing the growth rates of lattices and subgroups in simple Lie groups over local fields of positive characteristic, revealing their asymptotic equivalence.

Contribution

It provides the first proof of Lubotzky's conjecture on lattice growth rates in simple Lie groups over positive characteristic fields.

Findings

Growth rates of lattices and subgroups are asymptotically equivalent.

Number of maximal lattices with bounded covolume grows at a specific rate.

Confirmed conjecture for characteristic p > 5 cases.

Abstract

In this article we prove a conjecture of A. Lubotzky: let G = G_0(K), where K is a local field of characteristic p>5, G_0 is a simply connected, absolutely almost simple K-group of K-rank at least 2. We give the rate of growth of r_x(G) :={H< G| H a lattice in G; vol(G/H)<x}/~, where H_1\simH_2 if and only if there is an abstract automorphism t of G such that H_2 =t(H_1). We also study the rate of subgroup growth s_x(H) of any lattice H 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.