Cohomology and Mackey Functors for Profinite Groups

Monday 16th December 2019, Royal Holloway, University of London; to be held in Room 106-7 Senate House, Central London Local organiser: Brita Nucinkis

This will be the final meeting in 2019 of the Research Group Functor Categories for Groups (FCG).

All talks will take place in Room 106-7 of Senate House, Central London. The timetable is as follows:

  • 1:00-2:00pm: Nadia Mazza (Lancaster University), Mackey functors from finite to profinite groups
  • 2:15-3:15pm: Ged Corob Cook (University of the Basque Country), Probabilistic Finiteness Conditions for Profinite Groups
  • 3:15-3:45pm: Tea and coffee
  • 3:45-4:45pm: Thomas Weigel (Universita' di Milano-Bicocca), The fundamental group of a subfunctor of El for a pro-p group

Abstracts

Nadia Mazza (Lancaster University), Mackey functors from finite to profinite groups

In this talk, we review the different approaches to Mackey functors for finite groups, and propose a definition for profinite groups. We will discuss the Burnside Mackey functor in greater detail. This latter is ongoing work with Ilaria Castellano, Brita Nucinkis and Andrea Vera.

Ged Corob Cook (University of the Basque Country), Probabilistic Finiteness Conditions for Profinite Groups

A profinite group is said to be positively finitely generated if there is some n such that n random elements of the group (with respect to the Haar measure) generate it with positive probability. This turns out to be related to the growth of the number of maximal subgroups of a given index. I will show how to generalise this notion to the analogous conditions of positive finite presentation and positive type FPn, using the ideas of Frattini extensions and projective covers, and show how these conditions are related; positive type FPn corresponds to the growth of the sizes of cohomology groups with simple coefficients. Finally, we give some examples of groups displaying these behaviours, using the representation theory of finite simple groups.

Thomas Weigel (Universita' di Milano-Bicocca), The fundamental group of a subfunctor of El for a pro-p group Cohomological Mackey functors are a powerful tool to investigate the structure of an abstractly given profinite group G. For a pro-p group G, the cohomological Mackey functor El assigning every open subgroup U its elementary abelian quotient U/Φ(U), where Φ(U) denotes the Frattini subgroup of U, is of particular interest. Indeed, every subfunctor X ⊆ El is associated to a surjective homomorphism of pro-p groups π : G → π1(G,X) (and vice versa) and one may think of π1(-,X) as a kind of fundamental group of X. In the talk we present some elementary properties of this type of fundamental group, and indicate how one can use it for the study of ends of pro-p groups.