Pro-isomorphic zeta functions and p-adic integrals

Sun, 13/03/2016 - 12:00

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.


The computation of these Dirichlet series turns out to be equivalent to the computation of some p-adic integrals over algebraic groups; integrals of this type have been studied extensively since the 1960's.  Each of these two ways of approaching what turns out to be the same problem sheds light on the other.
The talk will discuss the connection between pro-isomorphic subgroups and p-adic integrals.  It will also discuss recent joint work with Mark Berman on the behavior of pro-isomorphic zeta functions under base extension.  No knowledge of the subject will be assumed.