New dimensional estimates for subvarieties of linear algebraic groups

TL;DR

This paper establishes new, explicit bounds on the intersection sizes of finite generating sets with subvarieties in linear algebraic groups, improving previous exponential-tower type bounds to more manageable exponential bounds.

Contribution

It provides the first general dimensional bounds with explicit exponential dependencies for subvarieties of linear algebraic groups over large fields.

Findings

Derived explicit bounds for intersections of generating sets with subvarieties.

Improved bounds from exponential-tower to exponential dependence on group dimension.

Immediate applications to diameter bounds in classical groups over finite fields.

Abstract

For every connected, almost simple linear algebraic group $G\leq\mathrm{GL}_{n}$ over a large enough field $K$, every subvariety $V\subseteq G$, and every finite generating set $A\subseteq G(K)$, we prove a general dimensional bound, that is, a bound of the form \[|A\cap V(\overline{K})|\leq C_{1}|A^{C_{2}}|^{\frac{\dim(V)}{\dim(G)}}\] with $C_{1},C_{2}$ depending only on $n,\mathrm{deg}(V)$. The dependence of $C_1$ on $n$ (or rather on $\dim (V)$) is doubly exponential, whereas $C_2$ (which is independent of $\mathrm{deg}(V)$) depends simply exponentially on $n$. Bounds of this form have proved useful in the study of growth in linear algebraic groups since 2005 (Helfgott) and, before then, in the study of subgroup structure (Larsen-Pink: $A$ a subgroup). In bounds for general $V$ and $G$ available before our work, the dependence of $C_1$ and $C_2$ on $n$ was of exponential-tower type. We draw immediate consequences regarding diameter bounds for untwisted classical groups $G(\mathbb{F}_{q})$. (In a separate paper, we derive stronger diameter bounds from stronger dimensional bounds we prove for specific families of varieties $V$.)