Extremal behavior of divisibility functions

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 $F_\Gamma(n) \preceq \log \log n$, then $\Gamma$ is finite. In contrast, when $\Gamma$ 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.