An approximate subgroup of a group $G$ is a symmetric subset $X\subset G$, such that the product set $XX$ is contained in a union of a finite number of $K$ translates of $X$. Groups lacking large approximate subgroups enjoy an expansion phenomenon, of interest in combinatorics and analysis; applying multiplication to any subset yields a substantially expanded set. Approximate subgroups have also made an appearance in model theory, for instance in the proof of Zilber's indecomposability theorem and its generalizations.

Using the pattern space of an appropriate local theory, we find a canonical locally compact space modelling, in the large, a given commensurability class of approximate subgroups. Using it we show that all approximate subgroups arise from a combination of homomorphisms into Lie groups, and the quasimorphisms of bounded cohomology.