14.00 – 14.45 : Naomi Andrew

(Laboratoire de Mathématiques d’Orsay)Title: Automorphisms behaving badly
Abstract: Baumslag–Solitar groups are HNN extensions of the infinite cyclic group, whose isomorphism type is controlled by two integers giving the two embeddings. They have provided many counterexamples over the years: for example, they include groups which are not Hopfian and groups which are Hopfian but not residually finite. Later, Collins and Levin showed that there are Baumslag–Solitar groups that do not have finitely generated automorphism group.

Moving this construction to higher rank, one can study « Leary–Minasyan groups »: these are HNN extensions of free abelian groups, with both inclusions finite index. They are also sources of counterexamples, such as groups which are CAT(0) but not biautomatic. We study their automorphism groups, and in particular characterise when they are finitely generated; this includes some finitely presented metabelian groups with automorphism groups that are not finitely generated. This is joint work with Sam Hughes and Motiejus Valiunas.

15.00 – 15.45 : Basile Morando (ENS – PSL) Title: On factoriality of the Neretin group von Neumann algebra
Abstract: To any locally compact group G, one can associate a von Neumann algebra L(G), generated by the image of G under its left regular representation. This algebra reflects decomposition properties of the representation: L(G) is a factor — i.e., has trivial center — if and only if the regular representation does not split as a direct sum of two disjoint subrepresentations.

In the discrete case, Murray and von Neumann showed in 1943 that L(G) is a factor if and only if all non-trivial conjugacy classes are infinite. By contrast, for non-discrete groups, determining factoriality becomes more subtle.

In this talk, we present a new sufficient criterion for factoriality of L(G), when G is a totally disconnected group. This criterion, based on a growth condition for the conjugation orbits of cosets, allows us to prove that the von Neumann algebra of the Neretin group is a factor — providing the first known example of a simple, non-discrete group with this property.

If time permits, we will also discuss implications of this criterion for determining the type of L(G), and for understanding factoriality of crossed product associated to G-actions on von Neumann algebras.

16.15 – 17.00 : Nicolas de Saxcé (CNRS & Université Paris-Nord)

Title: Approximation diophantienne et flots diagonaux dans les espaces de réseaux
Abstract: Dans un espace de réseaux on associe à toute orbite diagonale une suite d’éléments du groupe de Weyl satisfaisant certaines propriétés de convexité pour l’ordre de Bruhat, et qui décrit la position de l’orbite à distance finie près. Ce codage des orbites permet d’étudier l’approximation par des points rationnels dans les variétés de drapeaux.