Profinite and pro-p groups 2020/21

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?