GALS

The goal of this project is to study large surfaces from the geometric, topological and dynamical viewpoint. We interpret “large surfaces” in two ways: first, as families of surfaces for which some geometric characteristic goes to infinity, and second, as surfaces for which one of these characteristic is already infinite. Both subjects have recently attracted a lot of attention and spectacular results have been proven. The research community, though, is still relatively scattered and one of our ambitions is to bring together experts from various domains to create a common culture and a unifying approach to deal with questions in these topics. The project aims, on the one hand, to study hyperbolic and flat structures on surfaces with growing complexity. On the other hand, we are interested in surfaces of infinite topological type (or sometimes of infinite area). In particular, we will study algebraic and dynamical properties of mapping class groups of these surfaces, often from the viewpoint of geometric group theory (that is, via their actions on interesting spaces or by endowing them with a coarse geometric structure). Furthermore, we will be interested in closed geodesics on flat or hyperbolic surfaces of infinite-type, and we will look in particular at problems related to counting and distributions of these geodesics and questions concerning the length spectrum. Families of structures on infinite-type surfaces will also be at the center of this project: we will study the various notions of Teichmüller space for infinite-type hyperbolic surfaces, intersection functionals and currents, representations of their fundamental groups into higher rank Lie groups and perturbations of Anosov flows (Reeb flows) on them.