Intersection growth in groups

The intersection growth of a group $G$ is the asymptotic behavior of the index of the intersection of all subgroups of $G$ with index at most $n$, and measures the Hausdorff dimension of $G$ in profinite metrics. We study intersection growth in free groups and special linear groups and relate intersection growth to quantifying residual finiteness.
Comment: 20 pages, no figures. Revised version. We extend estimates to polycyclic groups and explain why normal intersection growth is a profinite invariant. Theorem 6.1 previously contained an error which has now been fixed