Gradients of Sequences of Subgroups in a Direct Product. (9th October 2017)

• Record Type: Journal Article
• Authors: Nikolov, Nikolay; Shemtov, Zvi; Shusterman, Mark
• Is Part Of: International mathematics research notices. Volume 2019:Number 12(2019)
• Page Start: 3704
• Page End: 3717
• Publication Date: 2017-10-09
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnx236
• Languages: English
• ISSNs: 1073-7928

Abstract

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