On Groups with Slow Intersection Growth

Intersection growth concerns the asymptotic behavior of the index of the intersection of all subgroups of a group that have index at most n. In this note we show that the intersection growth of some groups may not be a nicely behaved function by showing the following seemingly contradictory results: (a) for any group G the intersection growth function i_G(n) is super linear infinitely often; and (b) for any increasing function f there exists a group G such that i_G below f infinitely often.