Subgroup growth in free class-$2$-nilpotent groups

Abstract

We describe an effective procedure to compute the local subgroup zeta functions of the free class-$2$-nilpotent groups on $d$ generators, for all $d\ge 2$. For $d=4$, this yields a new, explicit formula. For $d\in \{4,5\}$, we compute the topological subgroup zeta functions. We also obtain general results about the reduced and topological subalgebra zeta functions. For the former, we determine the behaviour at one; for the latter, the degree and behaviours at zero and infinity. Some of these results confirm, in the relevant special cases, general conjectures by Rossmann.