| Résumé: |
A profinite group (pro-p group) is a type of compact
topological group that arises as a limit of finite groups (of order a
power of p), and shares many properties with finite groups (with
'compact' serving as a replacement for 'finite'). Such groups arise
naturally in several contexts, for instance as Galois groups, as
automorphism groups of rooted trees, or as algebraic groups over local
rings. To understand the difference between finite and profinite, it
is useful to consider those profinite groups which are 'minimal' among
infinite profinite groups. In finite group theory, the natural
minimal groups are the simple groups, that is, groups having no proper
normal subgroups (which is the same as having no proper subnormal
subgroups), and these have famously been classified. There are no
infinite simple profinite groups, but we can consider 'just infinite'
groups, that is, groups whose normal subgroups have finite index.
There is a stronger property of being 'hereditarily just infinite',
which says that every subgroup of finite index is itself just
infinite; equivalently, every subnormal subgroup has finite index.
Many 'naturally-occurring' profinite groups turn out to be just
infinite or have a large just infinite image. Current understanding
of just infinite profinite groups in general is much more limited than
it is for finite simple groups, with hereditarily just infinite groups
being especially mysterious, but there are some general results in
this area.
I will give a brief introduction to profinite groups and then discuss some general properties of (hereditarily) just infinite profinite groups (especially pro-p groups). |
| Résumé: | The notion of complete reducibility for spherical buildings was introduced by Serre. It is a natural generalisation of complete reducibility known from representation theory. In this context he formulated the center conjecture for spherical buildings whose proof was accomplished recently by Ramos-Cuevas, who managed to handle the most difficult case of $E_8$. In the first part of my talk I will explain complete reducibility and the center conjecture for spherical buildings. In the second part I will present a proof of the latter which uses the notion of a polar region. This proof is considerably shorter than the known proof, because it works for all types simultanously. It is based on a result of Timmesfeld and was obtained in joint work with Richard Weiss. |
| Résumé: | Il s'agit d'un travail en commun avec Amaury Thuillier et Annette Werner. Les immeubles euclidiens sont les analogues des espaces symétriques riemanniens ajustés au cas des groupes réductifs non archimédiens. Plus précisément, d'après F. Bruhat et J. Tits, on peut attacher à chaque groupe de ce type - par exemple SL(n,Q_p) - un complexe cellulaire admettant une action du groupe suffisamment transitive (et bien comprise) pour qu'elle permette d'en tirer des informations de structure très fines sur le groupe considéré. Le but du travail présenté est de proposer une technique de compactification qui est proche de celle qui avait été utilisée par I. Satake pour les espaces symétriques. Nous avions généralisé des idées de V. Berkovich au cas non déployé dans un premier travail ; il s'agira ici d'une variante utilisant des techniques de théorie des représentations. L'exposé comportera une grande première partie introductive. |
| Résumé: | This is joint work with Dan Segal (Oxford). An old question of Serre asks if the topology of finitely generated profinite group G is determined by its subgroups of finite index. A few years ago with Dan Segal we answered this affirmatively and in the process proved that many verbal subgroups of G are automatically closed, for example [G,G] or G^n. A natural extension of Serre's question is to find nontrivial restrictions on what kind of quotients can G have beyond the obvious continuous quotients. For example it is relatively easy to see that G cannot have Z or more generally a finitely generated infinite residually finite group as a quotient. In this talk I will present our recent results with Segal proving among other things that a compact Hausdorff group G cannot have an infinite finitely generated quotient. We also investigate the existence of dense normal subgroups of G. The main tool for proving this are results on finite groups of the following flavour: Let N be a normal subgroup of a finite group G. We prove that under certain (unavoidable) conditions the subgroup [N,G] is a product of commutators [N,y] (with prescribed values of y from a given set Y of G) of length bounded by a function of d(G) and |Y| only. Here d(G) is the minimal size of a generating set of G. |
| Résumé: | La fonction de Dehn D associée a une présentation de groupe <S, R> mesure la complexite du problème du mot pour cette présentation. Dans le cas où <S, R> est une présentation compacte d'un groupe localement compact G, le comportement asymptotique de D ne dépend que de la géometrie à grande échelle de G. En particulier, si G est un groupe algébrique sur un corps local et H est un réseau cocompact, alors D_G et D_H ont même comportement asymptotique. Dans un travail commun avec Yves de Cornulier, nous montrons que dans un groupe de Lie, D croît soit exponentiellement, soit polynomialement, et nous caractérisons algébriquement cette dichotomie. Par contre, dans le cas non-archimedien, nous montrons que si le groupe est compactement présenté, alors D croît au plus cubiquement. |
| Résumé: | I will begin by discussing a few elementary examples of group actions on buildings. The notion of strongly transitive actions (corresponding to BN-pairs) and Weyl-transitive actions (corresponding to Tits subgroups) will be defined, and I will discuss joint work with Ken Brown that provided the first concrete examples of Weyl-transitive actions that are not strongly transitive. The remainder of the time will be devoted to a discussion of recent joint work with Matthew Zaremsky, which provides many new examples of Weyl-transitive actions that are not strongly transitive. Though these examples cannot occur for spherical buildings, the analysis of yet another type of actions (which we call weakly transitive), crucial for establishing these examples for various affine buildings, might also be interesting in the spherical case. |
| Résumé: | In this self-contained lecture, we introduce and explain a connection between two-dimensional associative surjective algebras over arbitrary fields and certain (classical) varieties such as Hermitian Veronesean varieties, Segre varieties, and certain ring geometries such as Hjelmslev planes. We provide a common geometric construction and characterization. We motivate our research by proposing a magic 3x3 square, the left upper corner symbolizing the complex projective plane, and the right lower corner symbolizing the real E_6 building. |
| Résumé: | Prenant comme motivation un problème de théorie des groupes, je parlerai de matrices à coefficients entiers et du comportement asymptotique des suites vérifiant une relation de récurrence linéaire. |
| Résumé: | We give nice acute triangulations of the regular tetrahedron and the 3-cube. We show that for the 4-cube it is not possible, neither for R^4 if we assume that the tiles must have bounded geometry up to scaling. (Joint work with Eryk Kopczynski and Igor Pak.) |