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 $$ \lim_{n \to \infty} \frac{\log |\mathrm{Torsion}(U_n^{\mathrm{ab}})|}{[G : U_n]} = 0. $$
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2016
- DOI:
- 10.48550/arXiv.1609.08900
- arXiv:
- arXiv:1609.08900
- Bibcode:
- 2016arXiv160908900N
- Keywords:
-
- Mathematics - Group Theory;
- 20F05;
- 20F69;
- 20D05;
- 20F65;
- 37A20