Abstract

In his last paper Rufus Bowen, with Caroline Series defined a very particular map on the circle coming from Fuschian groups. Bowen was then able to prove, using that map, a result that became famous about Hausdorff dimension of "quasi-circles". The dimension result has been widely discussed but that was not the case for the construction of the map. Indeed, very few generalizations have been proposed and it was always under very restrictive hyperbolic geometry conditions. The goal of the talk is to describe the construction of similar maps, called Bowen-Series like maps, from combinatorial data, i.e. from some presentations of surface groups. The final object is a map satisfying nice dynamical properties that allow to get informations about the group, such as the "volume entropy" of the presentation, informations that are otherwise difficult to get from geometric group theory. The goal will be to discuss the constructions from a dynamical system point of view.