ESI Programme on
Profinite Groups

Overview

During recent years interest in profinite groups has grown. Profinite groups are used heavily during the investigation of questions in algebraic geometry, in algebraic number theory, but have also applications outside pure mathematics, for example in computer science (formal language theory) and in theoretical physics (renormalization theory). During the present meeting, however, we rather want to focus on developments within the field of profinite groups itself -- well understanding that many questions about such groups originate from other areas of mathematics.

Programme

It is suggested to have a conference on profinite groups during the first week, starting at 8 th of December 2008, while focussing at questions and problems in profinite group theory and the interaction between various researchers during the second week.

Some themes during the workshop

A rough outline of themes of international research, also to be considered during the ESI-workshop might be within the scope of the subsequent list (naturally not a ranking according to importance):

• Profinite completions. Every group can be eqipped with the profinite topology, turning it into a precompact totally disonnected (not necessarily Hausdorff) group. The residually finite groups are precisely those which become Hausdorff. Dan Segal and Nikolay Nikolov have proved that every topologically finitely generated profinite group has only subgroups of finite index that are closed. The heart of their argument is showing that the algebraic commutator subgroup is closed. Groups may well admit coarser topologies, for example arithmetic groups are often considered to be equipped with the congruence subgroup topology. Studying the congruence kernel, i.e., the kernel of the natural epimorphism of the profinite completion to the congruence completion of an arithmetic group is of great interest. Progress has come also from better understanding the structure of profinite groups acting on profinite trees, which has led to a Kurosh subgroup theorem for normal subgroups of profinite groups.
• (Co)homological methods. Profinite (co)homology is a tool in studying the possible structure of certain profinite Galois groups. Progress has been made in describing mainly profinite groups of low cohomological dimension.
• Combinatorial profinite and pro-p group theory. This area is concerned with profinite versions of the Kurosh-subgroup theorem, more generally -- Bass-Serre theory of profinite groups acting on profinite trees. Several analogies to what has been done in the discrete situation (along chapters of the book by Magnus-Karrass-Solitar and/or Lyndon Schupp) have been produced and raise interesting questions.
• Growth. Enormous work has been accomplished (Dan Segal, Alexander Lubotzky and several others) in understanding subgroup growth, namely the asymptotic behaviour of the number of subgroups of finite index in a given finitely generated group. These questions have been formulated and studied in great detail (mostly by the same authors) for profinite groups too. Today, representation growth and the extent of its influence on the structure of a group, is of great interest. More is known about subgroup growth for residually finite groups and for for profinite groups, particularly those appearing as profinite completions of residually finite ones.
• Just infinite groups. These groups are better known as tall groups in harmonic analysis. By definition such groups can only have finite quotients. Their understanding is still low but there are quite numerous examples of different flavour.

List of participants

Miklos ABERT (Chicago)	Menny AKA (Jerusalem)
Jorge ALMEIDA (Porto)	Karl AUINGER (Wien)
Yiftach BARNEA (London)	Nigel BOSTON (Madison)
Lior BARY-SOROKER (Tel Aviv)	Gunther BERGAUER (Wien)
Marco BOGGI (Paris)	Zoe CHATZIDAKIS (Paris)
Ido EFRAT (Be'er Sheva)	Liad FIREMAN (Jerusalem)
Fritz GRUNEWALD (Düsseldorf)	Dan HARAN (Tel Aviv)
Wolfgang HERFORT (Vienna)	Wolfram HOJKA (Wien)
Moshe JARDEN (Tel Aviv)	Otto H. KEGEL (U Freiburg i Br)
Helmut KOCH (HU Berlin)	Dessislava KOCHLOUKOVA (Campinas)
Jochen KOENIGSMANN (Freiburg i Br)	Pierre LOCHAK (Paris)
Karl LORENSEN (Altoona)	Alex LUBOTZKY (Jerusalem)
Andrea LUCCHINI (Padova)	Avinoam MANN (Jerusalem)
Chen MEIRI (Jerusalem)	Primoz MORAVEC (Ljubljana)
Aline G.S. PINTO (Brasilia)	Peter PLAUMANN (Erlangen)
Andrei RAPINCHUK (Virginia)	Luis RIBES (Ottawa)
Liudmila SABININA (Cuernavaca)	Dan SEGAL (Oxford)
Jean-Pierre SERRE (Paris)	Aner SHALEV (Jerusalem)
Benjamin STEINBERG (Ottawa)	Peter SYMONDS (Manchester)
Thomas WEIGEL (Milano Bicocca)	John S. WILSON (Oxford)
Reinhard WINKLER (Wien)	Pavel SHUMYATSKY (Brasilia)
Pavel ZALESSKII (Brasilia)	Theo ZAPATA (Brasilia)
Flavia F R ZAPATA (Brasilia)	Efim ZELMANOV (San Diego)