Brendan Murphy

Approximate subgroups in soluble linear groups over finite fields

Structure theorems for "approximate groups" show what structure remains when we relax the closure condition for groups. Examples of approximate groups are arithmetic progressions and large subsets of finite groups; a structure theorem shows how an arbitrary approximate group is composed of such examples. These results have applications to the construction of families of expander graphs, sieving in orbits of arithmetic groups, and spectral gap estimates in infinite volume hyperbolic surfaces.

In joint work with James Wheeler, we show that if a subset $A$ of $GL_n(\mathbb{F}_q)$ is "approximately closed" under multiplication and the group $G$ it generates is soluble, then there are subgroups $U$ and $S$ of $G$ such that:
$A$ quickly generates $U$,
$S$ contains most of $A$
$S/U$ is nilpotent.
Briefly: approximate soluble linear groups are (almost) finite by nilpotent. This confirms a conjecture of Helfgott.

The proof uses a sum-product theorem and exponential sum estimates, as well as some representation theory, but the talk will be mostly self-contained.