Uniform Character Bounds for Finite Classical Groups

TL;DR

This paper establishes exponential bounds on character ratios for finite classical groups, leading to significant applications such as confirming Thompson's conjecture for large simple groups of certain types.

Contribution

It provides the first exponential bounds on character ratios for all finite classical groups, with applications to longstanding conjectures.

Findings

Proved exponential bounds for character ratios in finite classical groups.

Confirmed Thompson's conjecture for large simple symplectic and certain orthogonal groups.

Extended bounds to orthogonal and unitary groups in characteristic 2.

Abstract

For every finite quasisimple group of Lie type $G$, every irreducible character $\chi$ of $G$, and every element $g$ of $G$, we give an exponential upper bound for the character ratio $|\chi(g)|/\chi(1)$ with exponent linear in $\log_{|G|} |g^G|$, or, equivalently, in the ratio of the support of $g$ to the rank of $G$. We give several applications, including a proof of Thompson's conjecture for all sufficiently large simple symplectic groups, orthogonal groups in characteristic $2$, and some other infinite families of orthogonal and unitary groups