 The structure of approximate groups by Emmanuel Breuillard and others

Published article: pmihes.centre-mersenne.org/articles/10.1007/s10240-012-0043-9/

Author-hosted full text: www.imo.universite-paris-saclay.fr/~emmanuel.breuillard/structure.pdf

Theorem 2.10 gives the approximate-group structure theorem. Corollary 11.9 and its proof, pages 68–69 in the inspected author version, obtain a bounded-index finite-by-nilpotent structure from one-scale relative polynomial growth. The proof uses a smaller scale with bounded growth; fixed scale constants may be adjusted by increasing the threshold.