Wednesday 1 February 2017 h.14:30, Room 2BC30

Leone Cimetta (Padova, Dip. Mat.)

“Zeta functions associated to profinite groups”

Abstract

In this seminar we will discuss the properties of some Dirichlet series associated to a group G satisfying specific topological properties. These series deal with two important problems arisen in the last century, which both had a great development over the last decades.

The first problem (the subgroup growth of a profinite group G) involves the behaviour of the function a_n(G), that is the number of subgroups of G of index n.

The second problem consists in determining the probability that, randomly choosing n elements of a group, we get a generating set for the whole group.

The second problem, in particular, arises from a famous work by P. Hall, which solved it in 1936 in the finite case.

After recalling some basic definitions, we will present the motivations for the problems; then, starting from some examples and classical results for finite groups, we will give some ideas to develop both problems in the profinite case and show some relations between the series involved.