Growth in linear groups

TL;DR

This paper proves a conjecture about the structure of sets with small tripling in linear groups, revealing subgroup configurations and implications for group diameters, advancing understanding of growth in linear algebraic groups.

Contribution

It establishes the structure of bounded tripling sets in linear groups, generalizing the Product Theorem and providing new bounds on diameters of quasirandom linear groups.

Findings

Sets with small tripling are covered by few cosets of a subgroup with nilpotent quotient.

The structure theorem includes the Product Theorem for finite simple groups of bounded rank.

Diameter of quasirandom finite linear groups is poly-logarithmic.

Abstract

We prove a conjecture of Helfgott on the structure of sets of bounded tripling in bounded rank, which states the following. Let $A$ be a finite symmetric subset of $\mathrm{GL}_n(\mathbf{F})$ for any field $\mathbf{F}$ such that $|A^3| \leq K|A|$. Then there are subgroups $H \trianglelefteq \Gamma \trianglelefteq \langle A \rangle$ such that $A$ is covered by $K^{O_n(1)}$ cosets of $\Gamma$, $\Gamma/H$ is nilpotent of step at most $n-1$, and $H$ is contained in $A^{O_n(1)}$. This theorem includes the Product Theorem for finite simple groups of bounded rank as a special case. As an application of our methods we also show that the diameter of sufficiently quasirandom finite linear groups is poly-logarithmic.