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 $L$.