Stability, corners, and other 2-dimensional shapes

We introduce a relaxation of stability, called robust stability, which is insensitive to perturbations by subsets of Loeb measure 0 in a pseudofinite group. We show that robust stability satisfies a stationarity principle for measure independent elements. We apply this principle to deduce the existence of squares and L-shapes in dense sub- sets of Cartesian squares of pseudofinite groups, possibly non-abelian. Our results imply qualitative asymptotic versions for Cartesian squares of finite groups.