Approximate subgroups of linear groups

TL;DR

This paper investigates the structure of approximate subgroups in linear groups like SL_n(k), showing they are either very small or nearly the entire group, with implications for expansion properties of Cayley graphs.

Contribution

The authors generalize Helfgott's results to higher dimensions and other algebraic groups, establishing new structural theorems for approximate subgroups in linear groups.

Findings

Approximate subgroups in SL_n(F_q) are either very small or nearly the whole group.

The results extend to other absolutely almost simple algebraic groups over finite fields.

Applications to expansion properties of Cayley graphs are forthcoming.

Abstract

We establish various results on the structure of approximate subgroups in linear groups such as SL_n(k) that were previously announced by the authors. For example, generalising a result of Helfgott (who handled the cases n = 2 and 3), we show that any approximate subgroup of SL_n(F_q) which generates the group must be either very small or else nearly all of SL_n(F_q). The argument generalises to other absolutely almost simple connected (and non-commutative) algebraic groups G over a finite field k. In a subsequent paper, we will give applications of this result to the expansion properties of Cayley graphs.