Profinite and pro-p groups 2020/21

Math PhD School Unimib-Unipv-INdAM

In the picture above, the Bicocca degli Arcimboldi (now owned by Pirelli): namely, the Renaissance villa (once in the countryland north to Milan) built by the Arcimboldi's family (literally, in Italian, «bicocca» means «little castle on the top of a hill»). The villa gave the name to the neighborhood, and to the eponymous University of Milano-Bicocca, which was built next to it.

OVERVIEW:

The course «Profinite and pro-p groups» was delivered in Winter 2020/21, as a course of the Joint PhD School in Mathematics Milano-Bicocca - Pavia - INdAM.

The course aims to introduce young researchers to the theory of profinite and pro-p groups (where p denotes a prime number). Profinite groups originated from Galois theory, as every Galois group is profinite, yet profinite and pro-p groups show up in several contexts. In particular, the impetus for current research on pro-p groups could be broadly described as coming from four directions: number theory; the problem of classifying finite p-groups; the theory of infinite groups; profinite group theory “in its own right”. The goal of the course is to give a comprehensive background on profinite and pro-p groups, and some examples of how the study of these groups may lead to results in the aforementioned branches of Algebra.

Abstract:

Preliminaries on topological groups and infinite Galois theory.
Profinite groups: definition, properties, examples.
Pro-p groups: definition, properties, examples.
Pro-p groups of finite rank.
Cohomology of profinite and pro-p groups.
Golod-Shafarevich inequality and Hilbert’s Class Field Tower Problem.
Further topics (depending on the audience’s requests): subgroup growth; pro-p groups and trees; pro- and finite- p-groups; congruence subgroups; pro-p Galois groups.

At the end of the course, each participant gives a talk on a topic related to the course, as a part of the final exam.

BIBLIOGRAPHY:
Here are the notes of the course, taken and typeset in LaTeX by Lorenzo Stefanello (whom I thank gratefully!). Although I do not doubt they were taken and written very carefully, I have not reviewed the notes yet, so you may consult them at your own risk! (Still I believe much of them are sound.) Here is the collection of the lavagne (whiteboards) I wrote during the lectures - again, at your own risk!

The main references used for the course are:

L. Ribes, Introduction to profinite groups, Travaux mathématiques. Vol. XXII, Trav. Math 22, Fac. Sci. Technol. Commun. Univ. Luxembourg 2013, link.
J.D. Dixon, M.P.F. du Sautoy, A. Mann and D. Segal, Analytic pro-p groups, 2nd ed., Cambridge Studies in Adv. Math. 61, CUP 1999, link.

Other references:

B. Klopsch, N. Nikolov and C. Voll, Lectures on profinite topics in group theory, LMS Student Texts 77, CUP, 2011, link.
L. Ribes and P.A. Zalesskiĭ, Profinite groups, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 40, Springer 2010, link.
M.P.F. du Sautoy, D. Segal and A. Shalev (eds.), New horizons in pro-p groups, Progress in Mathematics 184, Birkhäuser 2000, link.
A. Lubotzky and D. Segal, Subgroup growth, Progress in Mathematics 212. Birkhäuser 2003, link.

A COURSE-BOOK?

While preparing the material for the course, I couldn't help noticing that - as fas as I know, of course! - there are no books for an introductory - and comprehensive - course on profinite and profinite groups: indeed there are several good books on these topics, but either they are not suitable for graduate courses (such as Ribes-Zalesskiĭ's book), or they focus only on few aspects (for example, there is no book dealing with: structural aspects of pro-p groups, cohomology of pro-p groups, Galois pro-p groups, pro-p groups arising from acrions on trees... all topics that - I deem - should be mentioned in an introductory course on pro-p groups, and which I tried to menton in mine). So, if this gap will not be filled in the future, I would love to (try to) write such a comprehensive course-book, possibly not alone... anybody interested in?