Extremal behavior of divisibility functions

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