Résumé: | Kummer subspaces of tensor products of cyclic algebras, and in particular monomial ones, have gained popularity recently after being used for bounding the symbol length of central simple algebras over CM-fields. In this talk we shall focus mainly on the question of maximality and classification of such spaces, mainly in case of tensor products of cyclic algebras of degree 3, where the monomial spaces can be described in terms of graph theory. |
Résumé: | Given a finitely presented group Q, we produce a short exact sequence 1 → N → G → Q → 1 such that G is a torsion-free Gromov hyperbolic group without the unique product property and N is 2-generated. Varying Q, we obtain a wide diversity of concrete examples of Gromov hyperbolic groups without the unique product property. Our proof combines the famous Rips construction and the construction of Rips-Segev of groups without the unique product property. We explain those constructions using the graphical small cancellation theory over the free product. Finally, we further extend our construction to obtain the first Gromov hyperbolic groups without the unique product property and with Property (T). This is a joint work with Goulnara Arzhantseva. |
Résumé: | We describe necessary and sufficient conditions on a finite flag complex L ensuring the uniqueness, up to isomorphism, of a CAT(0) cube complex such that the link of every vertex is isomorphic to L. |
Résumé: | Un groupe libre peut-il agir proprement discontinûment sur R^{3} par transformations affines? Voici 30 ans, Margulis a montré que oui. J'exposerai une façon de paramétriser l'ensemble des exemples possibles par un objet combinatoire, le complexe des arcs d'une surface. Il s'agit d'un travail commun avec F. Kassel et J. Danciger. |
Résumé: | Both the class of Right-Angled Artin groups, and the Nielsen realisation problem, will be introduced. Then some recent progress towards solving the problem will be reported. |
Résumé: | An almost automorphism of a tree is a bijection on the vertices that preserves all but finitely many edge relations. This talk will present a canonical form for almost automorphisms that extends the classification of the automorphisms of a tree given by J. Tits in~1970. As Tits showed, every automorphism of a tree T either has finite orbits, in which case it fixes a vertex or is an inversion of an edge, or has infinite orbits, in which case it is a translation along a unique axis. In comparison, each almost automorphism, x of T, is the product of a part with finite orbits, called elliptic, and a part with infinite orbits, called hyperbolic. The hyperbolic part has a quasi-axis, which is a finite set of repelling and attracting ends of T together with a finite graph, G(x), that encodes how the almost automorphism transports vertices from the repelling to the attracting ends. The canonical form characterises the hyperbolic part of x up to conjugation in the totally disconnected, locally compact group, AAut(T), of all almost automorphisms of the tree. The quasi-axis of an almost automorphism x corresponds to compact, open subgroups of AAut(T) that are tidy for x. Indeed, the description of the quasi-axis comes about by following the tidying procedure that finds a tidy subgroup. The scale of x may then be computed in terms of G(x). |
Résumé: | A Bruhat-Tits building is an affine building whose building at infinity is Moufang. In his book The structure of affine buildings R. Weiss provided a detailed classification of Bruhat-Tits buildings of rank at least 3 which refines and extends Tits' classification of affine buildings of rank at least 4 considerably. In this work he also detemined the local structure all Bruhat-Tits buildings in most cases. The case of exceptional quadrangles turned out to be too involved. My talk is about joint work with H. Petersson and R. Weiss in which this case has been settled. It is based on the theory of Galois-descent and Tits-indices for affine buildings. |
Résumé: | Recently, a combinatorial approach have been developed to study quasiconformal structures of boundaries of hyperbolic groups. The motivation of this is to be able to characterize the regularity of a boundary that does not come with a good measure. Some how this can be done thanks to the Combinatorial Loewner Property (CLP). The CLP is a discrete version of the Loewner Property introduced by Heinonen and Koskela in abstract metric measured spaces. The CLP have been used for instance by M. Bourdon and B. Kleiner to give a new proof of the Cannon conjecture for Coxeter groups. In this talk I will give an overview of this theory and present an example of right-angled hyperbolic building of dimension 3 whose boundary satisfy the CLP. |
Résumé: | In 1988, Lubotzky, Philips and Sarnak have constructed optimal expanding graphs (i.e. Ramanujan graphs); beside being supreme expanders, these graphs also have high girth and high chromatic number. The construction was generalized in 2005 to simplicial complexes by Lubotzky, Samuels and Vishne, to produce the so called Ramanujan complexes. Both construction were based on translating the spectral properties of the graph or simplicial complex to properties of a certain representation of PGL_{d}(F) with F a local field. I will show that this principle is actually true in a much broader context, namely: Given any simplicial complex X and a group G acting on X, spectral properties of quotients of X can be expressed in terms of a certain representation of G. In particular, being Ramanujan corresponds to being weakly contained in the regular representation. As a result, one can derive further spectral properties of the complexes constructed by Lubotzky-Samuels-Vishne. |
Résumé: | We consider a locally compact group G and a G-space E. An irreducible probability measure μ on G is said to have the Liouville property on E if the G-invariant functions on E are the only continuous bounded functions on E that satisfy the mean value property with respect to μ. We will mainly discuss linear and projective linear actions. |
Résumé: | The Lyons group Ly is one of the 26 sporadic finite simple groups. It has order ≈ 5.10^{16} and is characterized by the property that the centralizer of an involution is a non-split extension of the alternating group of degree 11. This group can be described as the automorphism group of a rank 3 geometry Δ with diagram ˜G_{2}. The universal cover of Δ is a two-dimensional Euclidean building of the same type. It is one of the few examples for a non-classical affine building with a chamber-transitive automorphism group and therefore of special interest. |
Résumé: | A major theme in geometric group theory is to classify finitely generated groups up to quasi-isometry. We investigate the quasi-isometry classification of certain right-angled Coxeter groups. We first consider divergence, which was formulated as a quasi-isometry invariant by Gersten. Roughly speaking, this invariant measures how fast geodesic rays in W move away from each other. For W hyperbolic, the divergence of W is exponential, so we obtain a finer classification using topological features of the visual boundary. Combined with a theorem of Crisp and Paoluzzi, our results show that for right-angled Coxeter groups, quasi-isometry is strictly stronger than commensurability. This is joint work with Pallavi Dani. |
Résumé: | Various problems from group theory, geometry and combinatorics motivate the study of the natural action of the group Aut(F_{n}) on the set Epi(F_{n},G) of generating n-tuples in a group G generated by at least n elements. This action was mainly studied in the case when G is finite. We shall consider the situation when G is infinite and after an introduction into the subject we will discuss some new results on transitivity and nonamenability of this action. This is joint work with T. Nagnibeda. |
Résumé: | En équivalence orbitale, on étudie les actions de groupes en retenant uniquement la partition de l'espace en orbites induite par l'action, autrement dit en retenant la relation d'équivalence "être dans la même orbite". Dans le cadre des actions mesurées préservant une mesure de probabilités, il se trouve que cette relation d'équivalence est entièrement capturée par un groupe topologique, le groupe plein de la relation d'équivalence. Les propriétés du groupe plein sont alors autant d'invariants d'équivalence orbitale. On s'intéressera ici au rang topologique du groupe plein, i.e. le nombre minimal d'éléments nécessaires pour engendrer un sous groupe dense. On verra qu'il est très étroitement relié à un invariant fondamental d'équivalence orbitale: le coût. |
Résumé: | It is well known that most local results in Lie theory apply equally well to connected and p-adic locally compact Lie groups. However, many of the global theorems for connected Lie groups are not known to have p-adic counterparts. We present a global decomposition theorem for locally compact second countable p-adic Lie groups which is analogous to the solvable by semi-simple decomposition for connected locally compact Lie groups. |