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