On the character variety of the three-holed projective plane

We study the (relative) SL(2,C) character varieties of the three-holed projective plane and the action of the mapping class group on them. We describe a domain of discontinuity for this action, which strictly contains the set of primitive stable representations defined by Minsky, and also the set of convex-cocompact characters. We consider the relationship with the previous work of the authors and S. P. Tan on the character variety of the four-holed sphere.Comment: 27 page

INTRODUCTION TO POSITIVE REPRESENTATIONS AND FOCK-GONCHAROV COORDINATES

In this notes, we will try to give a simple description of the set of positive representations of the fundamental group of a surface with non-empty boundary to the group PSL m (R), as defined by Fock and Goncharov [1]. The construction uses a special set of coordinates on the space of all representations in PGL m (C) now called the Fock-Goncharov coordinates. For each step of the construction, we will consider the classical cases m = 2 and m = 3 before turning to the general case. We will also give the main property of these representations, namely their faithfulness and discreteness

Drawing disconnected graphs on the Klein bottle

We prove that two disjoint graphs must always be drawn separately on the Klein bottle, in order to minimize the crossing number of the whole drawing.Comment: 13 pages, second version, major changes in the proo

Dynamique de l’action du groupe modulaire et triplets de Markov

International audienceSoit $S$ une surface compacte avec $\chi (S) \le -1$. Nous nous intéressons ici à l’action du groupe modulaire de la surface $S$ sur les variétés de caractères $\mathcal{X} (\pi _1 (S) , \mathrm{SL}(2,\mathbb{C}) )$, lorsque $S$ est un tore à un trou ou une sphère à quatre trous. Le but de cet article est de présenter un objet combinatoire appelé application de Markov qui nous permet de définir un domaine de discontinuité ouvert pour l’action du groupe modulaire. L’intersection de ce domaine avec l’ensemble des caractères réels permet de retrouver certains résultats obtenus par Goldman dans le cas du tore à un trou et de montrer certains comportements nouveaux dans le cas de la sphère à quatre trous

On the character variety of the four-holed sphere

45 pages, 1 figureInternational audienceWe study the (relative) SL(2,C) character varieties of the four-holed sphere and the action of the mapping class group on it. We describe a domain of discontinuity for this action, and, in the case of real characters, show that this domain of discontinuity may be non-empty on the components where the relative euler class is non-maximal