Gradients of sequences of subgroups in a direct product

Journal article

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

Authors: Nikolov, N; Shemtov, Z; Shusterman, M

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Oxford University Press
• Journal: International Mathematics Research Notices
• Volume: 2017
• Pages: 1-14
• Publication date: 2017-10-09
• Acceptance date: 2017-09-11
• DOI: 10.1093/imrn/rnx236
• Keywords: math.GR