Workshop « Profinite groups and topology »
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.
&
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 »
