Résumé: | In this talk I present my recent work on locally compact C*-simple groups. A locally compact group is called C*-simple if its reduced group C*-algebra is simple, which has a representation theoretic interpretation. Work of Kalantar-Kennedy, Breuillard-Kalantar-Kennedy-Ozawa, Le Boudec, Kennedy and Haagerup gave a satisfactory shape to the theory of discrete C*-simple groups. This success motivates research on locally compact C*-simple groups. It turns out that every C*-simple group is totally disconnected: that fact will prescribe the bigger context of my talk. In particular, I will demonstrate how to exploit special features of operator algebras associated with totally disconnected groups and I will present natural questions in totally disconnected group theory arising from my work on C*-simplicity. |
Résumé: | Il est bien connu que toute variété riemannienne homogène simplement connexe M a la propriété de "rigidité locale-globale" suivante: étant donnée une variété simplement connexe N dont les boules de rayon 1 sont isométriques à la boule de rayon de 1 de M, alors N est isométrique à M. Dans un travail commun avec Mikael de la Salle, nous étudions une propriété similaire dans le cas d'objects singuliers comme les graphes de Cayley de réseaux dans les groupes de Lie simples et les immeubles affines de Bruhat–Tits. |
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The talk is about lattices on Ã_{2} buildings that preserve types and act regularly on panels of each type. We concentrate on the case where each vertex stabilizer acts on its residue as a Singer cycle. Jan Essert has shown that this class of lattices coincides with the class of groups admitting a presentation of a particular form.
For these lattices one would like to answer the following questions: • which of the lattices are isomorphic? • for which lattices is the building Bruhat–Tits? • for which of the lattices are the buildings isomorphic? • which of the lattices are commensurable to other known lattices (constructed by Köhler–Meixner–Wester, Ronan, Tits, and Cartwright–Mantero–Steger–Zappa)? I will report on recent progress in answering these questions, mostly restricted to the case where the branching parameter q is at most 5. |
Résumé: | Let G be a finitely generated group, and let dG be the word metric with respect to some finite generating set. let H be a subgroup of G. We say that H has bounded packing in G if for all R>0, there is an upper bound M(D) on the number of left cosets that are D-close. That is to say that if g_{1}H, … ,g_{M(D)}H are distinct left cosets, then there exists 1 ≤ i < j ≤ M(D) such that dG(g_{i}H,g_{j}H)>D. We prove the bounded packing property for any abelian subgroup of a group acting properly and cocompactly on a CAT(0) cube complex. The main ingredient of the proof is a cubical flat torus theorem. This is joint work with Dani Wise. |
Résumé: | A group action on a topological space X is said to be distal if for any two distinct points x,y ∈ X, the closure of the orbit of (x,y) in X × X does not intersect the diagonal set {(z,z) | z ∈ X}. I will describe some structural consequences of distal actions by conjugation inside t.d.l.c. groups. In particular, I will show that given a t.d.l.c. group G and a compactly generated subgroup H of G, then the conjugation action of H on G is distal if and only if it has small invariant neighbourhoods. |
Résumé: | Kac-Moody groups are natural generalisations of semisimple Lie groups (or better, of semisimple algebraic groups) to infinite dimension. In particular, they possess a (typically infinite-dimensional) Lie algebra, called a Kac-Moody algebra, which shares many properties with semisimple complex Lie algebras. The Lie correspondence between a Kac-Moody group and its Lie algebra, however, is far less understood than in the classical (finite-dimensional) setting. In this talk, I will present some aspects of this Lie correspondence, by explaining how certain maps between Kac-Moody algebras can be "exponentiated" to maps between the corresponding Kac-Moody groups. |
Résumé: | We describe explicit examples of countable groups acting on trees with trivial amenable radical and whose reduced C*-algebra is not simple. |
Résumé: | Anosov representations of hyperbolic groups in real reductive Lie groups are higher-rank generalizations of convex cocompact representations. I will explain how those provide proper actions on non-Riemannian homogeneous spaces. In certain cases, all proper actions of quasi-isometrically embedded groups arise from that construction. Joint work with F. Guéritaud, O. Guichard and A. Wienhard. |
Résumé: | Geometries of Coxeter type, as introduced by Tits in 1981, are geometries which are locally like buildings. These have applications in finite group theory, and in constructing buildings by passing to the universal cover. In 2014 Linus Kramer and Alexander Lytchak classified flag-transitive C_{3}-geometries with a compact topology such that the panels are connected. Such a geometry is either a building, is covered by building or is one of two exceptional geometries. The latter geometries were discovered in the context of polar actions by Podestà and Thorbergsson. In the talk I will illustrate how buildings and geometries arise from polar actions. After this, I will focus on the exceptional geometries encountered in the aforementioned classification, and provide a common construction of these using composition algebras. This construction is then useful to calculate the full automorphism group and show that these geometries are simply connected, answering a question by Kramer and Lytchak. |