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Géométrie et analyse sur les groupes discrets

Titel Englisch Geometry and analysis on discrete groups
Gesuchsteller/in Valette Alain
Nummer 163417
Förderungsinstrument Projektförderung (Abt. I-III)
Forschungseinrichtung Institut de mathématiques Université de Neuchâtel
Hochschule Universität Neuenburg - NE
Hauptdisziplin Mathematik
Beginn/Ende 01.10.2015 - 30.09.2018
Bewilligter Betrag 398'422.00
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Keywords (7)

Coarse embeddings; Broadcasting problem; Baum-Connes conjecture; Box spaces; Expander graphs; Affine isometric actions; Euclidean compression

Lay Summary (Französisch)

Lead
Géométrie et analyse sur les groupes discrets
Lay summary

 La théorie des groupes est la formulation mathématique de la notion intuitive de symétrie. Les groupes sont des objets algébriques qui, du fait de la nature géométrique de la symétrie, font le pont entre l'algèbre et la géométrie. En fait, un groupe G porte une géométrie intrinsèque: il y a une famille de graphes associés à un groupe, les graphes de Cayley de G, qui permettent de visualiser la structure algébrique. 

 Quand il s'agit d'étudier des groupes infinis, on peut faire appel au paradigme standard en mathématique: approcher un objet infini par des objets finis. Ici, les objets finis sont les quotients finis de G. Une question qui sera fondamentale dans ce projet est la suivante: supposons qu'on nous donne un graphe de Cayley pour chaque quotient fini de G; que pouvons-nous en déduire sur G? Le cas le plus simple de la question est: si on se donne la famille des cycles de longueur n (pour tout n) et qu'on sait qu'elle provient d'un groupe infini, pouvons-nous affirmer que ce groupe est le groupe additif des entiers? (la réponse est oui, dans ce cas simple). L'étude des graphes de Cayley débouche sur la théorie géométrique des groupes, un des domaines les plus actifs des mathématiques contemporaines. 

 

Direktlink auf Lay Summary Letzte Aktualisierung: 26.09.2015

Verantw. Gesuchsteller/in und weitere Gesuchstellende

Mitarbeitende

Verbundene Projekte

Nummer Titel Start Förderungsinstrument
130435 Groupes sofiques: algèbre, analyse et dynamique 01.05.2010 Sinergia
149261 Groupes discrets, variétés riemanniennes, et géométrie métrique 01.10.2013 Projektförderung (Abt. I-III)

Abstract

Group theory is the mathematical formulation of the intuitive notion of symmetry. Groups are algebraic objects but, in view of the intrinsically geometric nature of symmetry, they naturally bridge between algebra and geometry. Note that a group G carries an inherent geometry: there is a class of graphs associated to G, the Cayley graphs of G, which help visualize the algebraic structure. When it comes to studying infinite groups, analytic methods (from functional analysis or representation theory) naturally come into play. In this context we may appeal to the standard paradigm of trying to approximate the infinite by the finite. In the case of an infinite group G, one possible interpretation of this paradigm is: how well is G approached by its finite quotients? One question we want to address is the following: assume we know a Cayley graph for each finite quotient of G (such a collection of graphs is a box space of G), what do we know about G? The simplest case of that question is: assume we are given the family of cycles of length n for every n and are told that this comes for some infinite group G, can we assert that G is the group of integers? (The answer is ``yes'' in this simple case!) This type of question has proved to be extremely fruitful, the study of Cayley graphs giving rise to one of the most active areas of pure mathematics today, geometric group theory. The study of box spaces allows even more structural information about the group to be encoded geometrically, thus promising stronger results.As well as providing information about the group, a box space can be used to construct examples of metric spaces with interesting properties. In the varied world of metric spaces, it is natural to use a familiar, well-studied object such as a group in order to build examples. The connection with algebraic or analytic properties of the group allows us to tailor the properties of the box space to our needs. In coarse geometry (the study of ``large-scale'' properties of metric spaces), box spaces have consistently appeared as examples and counter-examples to important conjectures. We plan to exploit these connections to build metric spaces with interesting embeddability properties.The above coarse-geometric notions are invaluable tools in the progress on the intriguing Baum--Connes conjecture, a deep conjectured connection between a topological object and an analytic object in K-theory that can be associated to a group. This conjecture plays an important role in the non-commutative geometry programme, and implies several other conjectures in topology, geometry, and functional analysis, notably the Novikov conjecture and the Kaplansky-Kadison conjecture. The SNF grant would allow us to build a team active along the 4 following directions:1) Box spaces of residually finite groups, as a bridge between geometric and asymptotic group theory; connections with expander graphs.2) Coarse embeddability into Banach spaces.3)Proper isometric actions on Hilbert spaces.4) K-theory and K-homology computations involving certain metabelian groups.There are 4 people to be hired on that grant:a) M. Thiebout DELABIE and Mrs Sanaz POOYA, PhD students, 2015-2018.b) Dr. David HUME, post-doctoral collaborator, 18 months (oct. 2015-march 2017).c) Dr. Aglaia MYROPOLSKA, post-doctoral collaborator, 18 months (april 2017- sept. 2018).
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