Intersection growth in groups

Abstract

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.

arXiv comment20 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