Relative cubulations and groups with a 2-sphere boundary

We introduce a new kind of action of a relatively hyperbolic group on a $\text{CAT}(0)$ cube complex, called a relatively geometric action. We provide an application to characterize finite-volume Kleinian groups in terms of actions on cube complexes, analogous to the results of Markovic and Haïssinsky in the closed case.

Coadjoint Orbits, Cocycles and Gravitational Wess–Zumino

About 30 years ago, in a joint work with L. Faddeev we introduced a geometric action on coadjoint orbits. This action, in particular, gives rise to a path integral formula for characters of the corresponding group [Formula: see text]. In this paper, we revisit this topic and observe that the geometric action is a 1-cocycle for the loop group [Formula: see text]. In the case of [Formula: see text] being a central extension, we construct Wess–Zumino (WZ) type terms and show that the cocycle property of the geometric action gives rise to a Polyakov–Wiegmann (PW) formula with boundary term given by the 2-cocycle which defines the central extension. In particular, we obtain a PW type formula for Polyakov’s gravitational WZ action. After quantization, this formula leads to an interesting bulk-boundary decoupling phenomenon previously observed in the WZW model. We explain that this decoupling is a general feature of the Wess–Zumino terms obtained from geometric actions, and that in this case, the path integral is expressed in terms of the 2-cocycle which defines the central extension. In memory of our teacher Ludwig Faddeev

Asymptotical Behaviour of Roots of Infinite Coxeter Groups

Let W be an infinite Coxeter group. We initiate the study of the set E of limit points of “normalized” roots (representing the directions of the roots) of W. We show that E is contained in the isotropic cone Q of the bilinear form B associated with a geometric representation, and we illustrate this property with numerous examples and pictures in rank 3 and 4. We also define a natural geometric action of W on E, and then we exhibit a countable subset of E, formed by limit points for the dihedral reflection subgroups of W. We explain how this subset is built fromthe intersection with Q of the lines passing through two positive roots, and finally we establish that it is dense in E.

Asymptotical behaviour of roots of infinite Coxeter groups I

International audience Let $W$ be an infinite Coxeter group, and $\Phi$ be the root system constructed from its geometric representation. We study the set $E$ of limit points of "normalized'' roots (representing the directions of the roots). We show that $E$ is contained in the isotropic cone $Q$ of the bilinear form associated to $W$, and illustrate this property with numerous examples and pictures in rank $3$ and $4$. We also define a natural geometric action of $W$ on $E$, for which $E$ is stable. Then we exhibit a countable subset $E_2$ of $E$, formed by limit points for the dihedral reflection subgroups of $W$; we explain how $E_2$ can be built from the intersection with $Q$ of the lines passing through two roots, and we establish that $E_2$ is dense in $E$. Soit $W$ un groupe de Coxeter infini, et $\Phi$ le système de racines construit à partir de sa représentation géométrique. Nous étudions l'ensemble $E$ des points d'accumulation des racines "normalisées'' (représentant les directions des racines). Nous montrons que $E$ est inclus dans le cône isotrope $Q$ de la forme bilinéaire associée à $W$, et nous illustrons cette propriété à l'aide de nombreux exemples et images en rang $3$ et $4$. Nous définissons une action géométrique naturelle de $W$ sur $E$, pour laquelle $E$ est stable. Puis nous présentons un sous-ensemble dénombrable $E_2$ de $E$, constitué des points d'accumulation associés aux sous-groupes de réflexion diédraux de $W$ ; nous expliquons comment $E$ peut être construit à partir des points d'intersection de $Q$ avec les droites passant par deux racines, et nous montrons que $E_2$ est dense dans $E$.

On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings

Given a complete CAT(0) space X endowed with a geometric action of a group Ⲅ, it is known that if Ⲅ contains a free abelian group of rank n, then X contains a geometric flat of dimension n. We prove the converse of this statement in the special case where X is a convex subcomplex of the CAT(0) realization of a Coxeter group W, and Ⲅ is a subgroup of W. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits buildings.

A NORMAL FORM FOR BRAIDS

We present a seemingly new normal form for braids, where every braid is expressed using a word in a regular language on some simple alphabet of elementary braids. This normal form stems from analysing the geometric action of braid groups on curves in a punctured disk.