Quantum frieze patterns in quantum cluster algebras of type A

We introduce a quantisation of the Coxeter-Conway frieze patterns and prove that they realise quantum cluster variables in quantum cluster algebras associated with linearly oriented Dynkin quivers of type A. As an application, we obtain the explicit polynomials arising from the lower bound phenomenon in these quantum cluster algebras.Comment: 10 page

Dynamics on nilpotent character varieties

Let R(N,G) be the connected component of the identity of the variety of representations of a finitely generated nilpotent group N into a connected compact Lie group G, and let X(N,G) be the corresponding moduli space. We show that there exists a natural Out(N)-invariant measure on X(N,G) and that whenever Out(N) has at least one hyperbolic element, the action of Out(N) on X(N,G) is mixing with respect to this measure.Comment: 17 pages, Version 2 has minor updates and corrections, accepted for publication in Conformal Geometry and Dynamic

Triangulation minimale de cubes

Une manière combinatoire d'étudier les triangulations de cubes n-dimensionnels est donnée, ainsi que des outils pour optimiser le nombre de simplexes utilisés dans une telle triangulation. On donne ensuite un exemple concret en démontrant la minimalité d'une triangulation du 3-cube

Generalizations of Schottky groups

Schottky groups are classical examples of free groups acting properly discontinuously on the complex projective line. We discuss two different applications of similar constructions. The first gives examples of 3-dimensional Lorentzian Kleinian groups which act properly discontinuously on an open dense subset of the Einstein universe. The second gives a large class of examples of free subgroups of automorphisms groups of partially cyclically ordered spaces. We show that for a certain cyclic order on the Shilov boundary of a Hermitian symmetric space, this construction corresponds exactly to representations of fundamental groups of surfaces with boundary which have maximal Toledo invariant

Triangulation minimale de cubes

Une manière combinatoire d'étudier les triangulations de cubes n-dimensionnels est donnée, ainsi que des outils pour optimiser le nombre de simplexes utilisés dans une telle triangulation. On donne ensuite un exemple concret en démontrant la minimalité d'une triangulation du 3-cube

Foliations between crooked planes in 3-dimensional Minkowski space

International audienc

Einstein tori and crooked surfaces

-Corrected typos -Added definitions and details to AdS crooked planes section -Clarified choice of representatives for Plucker coordinates in Section 4 -Added a figure of a crooked surfaceIn hyperbolic space, the angle of intersection and distance classify pairs of totally geodesic hyperplanes. A similar algebraic invariant classifies pairs of hyperplanes in the Einstein universe. In dimension 3, symplectic splittings of a 4-dimensional real symplectic vector space model Einstein hyperplanes and the invariant is a determinant. The classification contributes to a complete disjointness criterion for crooked surfaces in the 3-dimensional Einstein universe