On finite sets of small tripling or small alternation in arbitrary groups

We prove Bogolyubov-Ruzsa-type results for finite subsets of groups with small tripling, $\mathrm{\mid }{A}^3\mathrm{\mid }\le O(\mathrm{\mid }A\mathrm{\mid })$, or small alternation, $|AA^{$\text{-}$1} A|\leq O(|A|)$. As applications, we obtain a qualitative analog of Bogolyubov’s Lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result generalizes the abelian case, due to Alon, Fox, and Zhao, and gives a quantitative version of previous work of the author, Pillay, and Terry.