Workshop « Profinite groups and topology »

Marseilles

May 8th-11th, 2017

Marseilles

May 8th-11th, 2017

Speakers

Louis Funar (Institut Fourier, Grenoble)

Benoît Loisel (École polytechnique, Palaiseau)

Alex Lubotzky (Hebrew University Jerusalem)

Sylvain Maillot (IMAG, Montpellier)

Luisa Paoluzzi (I2M, Marseille)

T.N. Venkataramana (TIFR Mumbai)

Gareth Wilkes (Oxford)

Organizers

Michel Boileau (Aix-Marseille Université)

Peter Haïssinsky (Aix-Marseille Université)

Bertrand Rémy (École polytechnique, Palaiseau)

Location

The talks will be given at the following place:

F.R.U.M.A.M.

Bâtiment 7

Aix Marseille Université

Site St-Charles

3 place Victor Hugo

13003 Marseille

See more here in general and here for directions.

Schedule

Monday 8th

14h-15h15 Alex Lubotzky: Presentations of profinite groups

16h-17h15 Alex Lubotzky: Phantom finite subgroups of torsion free groups

Tuesday 9th

9h15-10h30 Venkataramana: Grothendieck closure of an integral linear group, 1

11h-12h15 Louis Funar: Representations of mapping class groups and applications, 1

14h30-15h45 Gareth Wilkes: Profinite completions of 3-manifold groups, 1

16h30-17h30 Benoît Loisel: Explicit minimal generation of some linear pro-p groups

Wednesday 10th

9h15-10h30 Venkataramana: Grothendieck closure of an integral linear group, 2

11h-12h15 Louis Funar: Representations of mapping class groups and applications, 2

14h30-15h45 Gareth Wilkes: Profinite completions of 3-manifold groups, 2

16h30-17h30 Luisa Paoluzzi: Character varieties of knots and their reductions mod p

Thursday 11th

9h15-10h30 Venkataramana: Grothendieck closure of an integral linear group, 3

11h-12h15 Louis Funar: Representations of mapping class groups and applications, 3

14h30-15h45 Gareth Wilkes: Profinite completions of 3-manifold groups, 3

16h30-17h30 Sylvain Maillot: Structure theory of open graph 3-manifolds

Abstracts

Louis Funar : « Representations of mapping class groups and applications »

Abstract: Topological quantum field theories developed by Reshetikhin and Turaev in dimension 3 provide families of projective representations of mapping class groups. Our aim is to present some partial results concerning the kernels and the images of a particular family issued from skein theory of links. As applications we derive a large supply of finite quotients of mapping class groups and Burnside-type surface groups.

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Benoit Loisel: « Explicit minimal generation of some linear pro-p groups »

Abstract : let G be a semisimple group defined over a non-Archimedean local field K, typically SLn(Q_p), with residual characteristic p. We have a topological group structure on the group of rational points G(K). The maximal compact subgroups of G(K) can be realized as some stabilizers for a suitable action of the group G(K) on a polysimplicial complex called the Bruhat-Tits building. In this presentation, we begin with an short introduction to the theory of Bruhat-Tits buildings, which is used to describe the maximal compact and pro-p subgroups of G(K). The latter play a role analogous to that of the p-Sylows of a finite group and are, in particular, pairwise conjugated. Under suitable assumptions, we can then explicitly describe a minimal set of topological generators of a maximal pro-p subgroup. The minimal number of these topological generators is then linear in some combinatorial datum defined from G, namely the rank of a well-chosen root system.

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Alex Lubotzky: « Presentations of profinite groups »

Abstract: we will present some basic results on the presentations of profinite groups as well as some application to presentations of discrete groups. Some questions for further research will also be presented.

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Alex Lubotzky: « Phantom finite subgroups of torsion free groups »

Abstract: There exists a finitely generated residually-finite torsion-free group whose profinite completion contains an isomorphic copy of every finite group.

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Sylvain Maillot: « Structure theory of open graph 3-manifolds »

Abstract: Motivated by some questions in Riemannian geometry, we introduce the class of noncompact graph 3-manifolds. We show that some of the structure theory of compact graph manifolds, due to Waldhausen in the late 60s, goes through. However, some results do not; we will present examples to that effect.

Part of this is still work in progress.

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Luisa Paoluzzi: « Character varieties of knots and their reductions mod p »

Abstract: I will present joint work with Joan Porti (UAB) in which the SL(2,C)-character varieties of certain knots are studied. A first class of examples concerns a family of Montesinos knots whose character varieties contain high-dimensional irreducible components, different from the Teichmüller one. A second class of exemples concerns knots admitting either periodic or free symmetries: in this case, the existence and type of symmetries is reflected in the structure of (part of) the character variety.

The interest of these varieties stems from the fact that they provide toy-examples of possible "ramification" phenomena that can occur when considering character varieties over algebraically closed fields of positive characteristic, instead of over the field C of complex numbers

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Venkataramana: « Grothendieck Closure of an integral linear group »

Abstract: a complete characterisation of integral linear groups whose Grothendieck closure is itself given. The criterion is in terms of the congruence subgroup property. As a consequence, we deduce that the free group on two generators is not equal to its Grothendieck closure, answering an old question in the affirmative.

This is joint work with Alex Lubotzky.

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Gareth Wilkes: « Profinite completions of 3-manifold groups »

Abstract: I will give a survey of the current results concerning the profinite completions of 3-manifold groups. Particular attention will be paid to "profinite rigidity" type results - that is, those results aiming to classify 3-manifold groups with the same profinite completion. The focus will be on describing general themes and introducing the audience to the various techniques which contribute to these results rather than detailed proofs. Familiarity with 3-manifold topology will be helpful but not necessary.

Financial support

Projet ANR 12-BS01-0003-01 GDSous/GSG « Géométrie des sous-groupes »