Gradients of sequences of subgroups in a direct product

Abstract

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 $$\underset{n\to \infty }{\lim }\frac{\min \{|X|:⟨X⟩=U_n\}}{[G:U_n]}=0$$ 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. For $A,B$ that are finitely presented we show that $$\underset{n\to \infty }{\lim }\frac{\log |\mathrm{Torsion}(U_n^{\mathrm{ab}})|}{[G:U_n]}=0.$$