Rank gradient of sequences of subgroups in a direct product

Nikolay Nikolov, Zvi Shemtov, Mark Shusterman

Published 2016-09-28Version 1

For a sequence $\{U_n\}_{n = 1}^\infty$ of finite index subgroups of a direct product $G = A \times B$ of finitely generated groups, we show that $$\lim_{n \to \infty} \frac{\min\{|X| : \langle X \rangle = U_n\}}{[G : U_n]} = 0$$ once $[A : A \cap U_n], [B : B \cap U_n] \to \infty$ as $n \to \infty$. Our proof relies on the classification of finite simple groups.