Subgroup Growth in Some Profinite Chevalley Groups

TL;DR

This paper improves bounds on subgroup growth of certain profinite Chevalley groups over power series fields by introducing a new parameter and deriving new algebraic estimates.

Contribution

It introduces the ridgeline number $v( extbf{G})$ and provides improved bounds for subgroup growth using new Lie algebra codimension estimates.

Findings

Established a new uniform bound for subgroup growth.

Introduced the ridgeline number $v( extbf{G})$ as a key parameter.

Derived a novel estimate for the codimension of $[U,V]$ in Lie algebras.

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 $\mathbf{G}$.