Characteristic Subgroup Growth

Abstract

Let $s_n^{\mathrm{ch}}(\hat{I}“)$ denote the number of characteristic subgroups of index at most $n$ in a finitely generated group $\hat{I}“$. In response to a question of I. Rivin we show that if $\hat{I}“=F_r$ is the free group on $r\ge 2$ generators then the growth type of $s_n^{\mathrm{ch}}(F_r)$ is $n^{\log (n)}$. This is in contrast with the expectation of W. Thurston who predicted that there should be a difference between $r=2$ and $r>2$. Along the way we answer a question of arXiv:1703.07866 on the normal subgroup growth of large groups.

8 pages