Solving equations in finite groups and complete amalgamation

Abstract: Roth's theorem on arithmetic progression states that a subset $A$ of the natural numbers of positive upper density contains an arithmetic progression of length 3, that is, the equation $x+z=2y$ has a solution in $A$.
Finitary versions of Roth's theorem study subsets $A$ of $\{0,\ldots, N\}$, and ask whether the same holds for sufficiently large $N$, for a fixed lower bound on the density. In a similar way, concerning finite groups, one may study whether or not sufficiently large sets of a finite group contain solutions of an equation, or even a system of equations. For instance, for the equation $x·y=z$, Gowers (2008) showed that any subset of a finite simple non-abelian group will contain many solutions to this equation, provided it has sufficiently large density.

In this talk I will report on recent work with Amador Martin-Pizarro on how to find solutions to the above equations in the context of pseudo-finite groups, using techniques from model theory which resonate with (a group version of) the independence theorem in simple theories due to Pillay, Scanlon and Wagner.