Sets preserved by a large subgroup of the special linear group

TL;DR

This paper investigates the structure of subsets in the affine plane over finite fields that are preserved by large subgroups of SL_2(F_q), establishing bounds and conditions under which such sets must be contained in a line.

Contribution

It provides bounds on the subgroup size preserving a set and characterizes sets preserved by many group elements as being contained in a line, using combinatorial and incidence geometry methods.

Findings

Bounds on subgroup size preserving a set in the affine plane.

Sets preserved by many group elements are contained in a line.

Results are sharp and rely on incidence bounds in finite fields.

Abstract

Let $E$ be a subset of the affine plane over a finite field $\mathbb{F}_q$. We bound the size of the subgroup of $SL_2(\mathbb{F}_q)$ that preserves $E$. As a consequence, we show that if $E$ has size $\ll q^\alpha$ and is preserved by $\gg q^\beta$ elements of $SL_2(\mathbb{F}_q)$ with $\beta\geq 3\alpha/2$, then $E$ is contained in a line. This result is sharp in general, and will be proved by using combinatorial arguments and applying a point-line incidence bound in $\mathbb{F}_q^3$ due to Mockenhaupt and Tao (2004).