Congruence subgroup growth of arithmetic groups in positive characteristic

Abstract

We prove a new uniform bound for subgroup growth of a Chevalley group $G$ over the local ring $\mathbb {F}[[t]]$ and also over local pro-$p$ rings of higher Krull dimension. This is applied to the determination of congruence subgroup growth of arithmetic groups over global fields of positive characteristic. In particular, we show that the subgroup growth of ${\rm SL}\sb n(F\sb p[t]) (n\geq3)$ is of type $n\sp {\log n}$. This was one of the main problems left open by A. Lubotzky in his article [5].

The essential tool for proving the results is the use of graded Lie algebras. We sharpen Lubotzky's bounds on subgroup growth via a result on subspaces of a Chevalley Lie algebra $L$ over a finite field $\mathbb {F}$. This theorem is proved by algebraic geometry and can be modified to obtain a lower bound on the codimension of proper Lie subalgebras of $L$.