Pro-isomorphic zeta functions and p-adic integrals
A finitely generated group $G$ has only a finite number, say $a_n(G)$, of subgroups of any given index $n$. The study of subgroup growth, i.e. of the behavior of this sequence, has been an active area of research for several decades. A variant problem investigates the sequence $a_n^\wedge (G)$ counting subgroups of index $n$ whose profinite completion is isomorphic to that of the original group $G$, and in particular the analytic properties of the Dirichlet series derived from this sequence.