Growth in linear groups

2021/07/14 by Sean Eberhard, Eberhard, Sean, Brendan Murphy +5
Mathematics · Computer Science · #Finite Group Theory Research #Limits and Structures in Graph Theory #Advanced Graph Theory Research

paper · doi:10.48550/arxiv.2107.06674

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 GL_n(F) for any field F such that |A³| ≤ K|A|. Then there are subgroups H \trianglelefteq Γ\trianglelefteq ⟨ A ⟩ such that A is covered by K^O_n(1) cosets of Γ, Γ/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.

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