On the Shortest Identity in Finite Simple Groups of Lie Type

TL;DR

This paper establishes bounds on the length of the shortest identity in finite simple groups of Lie type, providing explicit polynomial bounds in terms of the group's defining parameters.

Contribution

It offers the first explicit polynomial bounds on the shortest identity length in finite simple groups of Lie type, linking algebraic properties to combinatorial bounds.

Findings

Shortest identity length is bounded by explicit polynomials in q and r.

Provides both upper and lower bounds for the identity length.

Connects algebraic structure with combinatorial bounds in finite groups.

Abstract

We prove that the length of the shortest identity in a finite simple group of Lie type of rank $r$ defined over $\mathbb{F}_q$, is bounded (from above and below) by explicit polynomials in $q$ and $r$.