Extremal behavior of divisibility functions

Abstract

In this short article, we study the extremal behavior $F_{\Gamma }(n)$ of divisibility functions $D_{\Gamma }$ introduced by the first author for finitely generated groups $\Gamma$. We show finitely generated subgroups of $\GL(m,K)$ for an infinite field $K$ have at most polynomial growth for the function $F_{\Gamma }(n)$. Consequently, we obtain a dichotomy for the growth rate of $\log F_{\Gamma }(n)$ for finitely generated subgroups of $\GL(n,\C)$. We also show that if $F_{\Gamma }(n)⪯\log \log n$, then $\Gamma$ is finite. In contrast, when $\Gamma$ contains an element of infinite order, $\log n⪯F_{\Gamma }(n)$. We end with a brief discussion of some geometric motivation for this work.

10 pages (added Lemma 2.1 and some details to the main proof of Theorem 1.1 to address a gap pointed out by a referee)

References in corpus (2)

• Quantifying Residual Finiteness
• Approximating a group by its solvable quotients