Résumé : A marked surface S is a finite-type possibly non-orientable surface with finitely many marked points (P) on the boundary and in the interior. These surfaces have long been studied both from a combinatorial as well as a geometric perspective. When the Euler characteristic of S\P is negative, the surface admits a finite-area hyperbolic metric. Such surfaces are called crowned surfaces, because their boundaries resemble a crown with spikes. Associated to marked/crowned surfaces is a combinatorial and topological object called the arc complex. This is a simplicial complex generated by arcs whose endpoints lie on the marked points. The arc complex has been used to understand the geometry of the surface by various mathematicians like Penner, Harer, Bowditch, Epstein. The arc complex is almost always infinite. In this talk we will focus on four families of surfaces for which it is finite. We will discuss how the topology of this complex helps us to understand certain deformations of crowned surfaces that weakly increase the "distances between spikes". As a result we get the non-simple polytopal realisations of these finite simplicial complexes. This is based ont the joint work with François Guéritaud. https://arxiv.org/abs/2505.01285

 [arXiv]

Dernière modification : Friday 09 May 2025

Contact pour cette page : Cyril.Banderier at lipn.univ-paris13.fr