Counting lattices in simple Lie groups: The positive characteristic case

Abstract

In this article, we prove the following conjecture by Lubotzky. Let $G={\mathbb{G}}_0\left(K\right)$, where $K$ is a local field of characteristic $p\ge 5$ and where ${\mathbb{G}}_0$ is a simply connected, absolutely almost simple $K$-group of $K$-rank at least 2. We give the rate of growth of

$${\rho }_x\left(G\right):=|\{\Gamma \subseteq G|\Gamma \text{ a lattice in }G,\mathrm{vol}(G/\Gamma )\le x\}/\sim |,$$

where ${\Gamma }_1\sim {\Gamma }_2$ if and only if there is an abstract automorphism $\theta$ of $G$ such that ${\Gamma }_2=\theta \left({\Gamma }_1\right)$. We also study the rate of subgroup growth $s_x(\Gamma )$ of any lattice $\Gamma$ 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 $x$.