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Suppose $G$ is a finite group and $A\subseteq G$ is such that $\{gA:g\in G\}$has VC-dimension strictly less than $k$. We find algebraically well-structuredsets in $G$ which, up to a chosen $\epsilon>0$, describe the structure of $A$and behave regularly with respect to translates of $A$. For the subclass ofgroups with uniformly fixed finite exponent $r$, these algebraic objects arenormal subgroups with index bounded in terms of $k$, $r$, and $\epsilon$. Forarbitrary groups, we use Bohr neighborhoods of bounded rank and width insidenormal subgroups of bounded index. Our proofs are largely model theoretic, andheavily rely on a structural analysis of compactifications of pseudofinitegroups as inverse limits of Lie groups. The introduction of Bohr neighborhoodsinto the nonabelian setting uses model theoretic methods related to the work ofBreuillard, Green, and Tao and Hrushovski on approximate groups, as well as aresult of Alekseev, Glebskii, and Gordon on approximate homomorphisms.

Structure and regularity for subsets of groups with finite VC-dimension