Subgroup Growth in Some Profinite Chevalley Groups

Abstract

In this article we improve the known uniform bound for subgroup growth of Chevalley groups over $\mathbf{G}({\mathbb{F}}_p[[t]])$. We introduce a new parameter, the ridgeline number $v(\mathbf{G})$, and give new bounds for the subgroup growth of $\mathbf{G}({\mathbb{F}}_p[[t]])$ expressed through $v(\mathbf{G})$. We achieve this by deriving a new estimate for the codimension of $[U,V]$ where $U$ and $V$ are vector subspaces in the Lie algebra of $\mathbf{G}$.

11 pages; minor revisions throughout, the proof of Theorem 0.4 is reorganised