Growth in linear groups

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.