Gradients of sequences of subgroups in a direct product

TL;DR

This paper investigates the asymptotic behavior of subgroup generators and torsion in sequences of finite index subgroups within direct products of finitely generated groups, revealing that certain normalized measures tend to zero.

Contribution

It establishes new limits for the growth of minimal generating sets and torsion in subgroup sequences of direct product groups, utilizing the classification of finite simple groups.

Findings

Normalized minimal generating set size tends to zero.

Torsion growth in abelianizations is negligible asymptotically.

Results apply to sequences where subgroup intersections with factors grow unbounded.

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 $$ \lim_{n \to \infty} \frac{\log |\mathrm{Torsion}(U_n^{\mathrm{ab}})|}{[G : U_n]} = 0. $$