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

  • G. Conant

    University of Cambridge, UK
  • Anand Pillay

    University of Notre Dame, USA
  • C. Terry

    University of Chicago, USA
Structure and regularity for subsets of groups with finite VC-dimension cover
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Abstract

Suppose is a finite group and is such that has VC-dimension strictly less than . We find algebraically well-structured sets in which, up to a chosen , describe the structure of and behave regularly with respect to translates of . For the subclass of groups with uniformly fixed finite exponent , these algebraic objects are normal subgroups with index bounded in terms of , , and . For arbitrary groups, we use Bohr neighborhoods of bounded rank and width inside normal subgroups of bounded index. Our proofs are largely model-theoretic, and heavily rely on a structural analysis of compactifications of pseudofinite groups as inverse limits of Lie groups. The introduction of Bohr neighborhoods into the nonabelian setting uses model-theoretic methods related to the work of Breuillard, Green, and Tao [8] and Hrushovski [28] on approximate groups, as well as a result of Alekseev, Glebskiĭ, and Gordon [1] on approximate homomorphisms.

Cite this article

G. Conant, Anand Pillay, C. Terry, Structure and regularity for subsets of groups with finite VC-dimension. J. Eur. Math. Soc. 24 (2022), no. 2, pp. 583–621

DOI 10.4171/JEMS/1111