Geometrization of piecewise homeomorphisms of the circle

Abstract

In 1979 R. Bowen and C. Series gave a beautiful construction of a dynamical system associated to a Fuschian group. They constructed a particular map that was a piecewise diffeomorphism of the circle from the action of the group on the hyperbolic 2 space. More recently I generalized this idea to construct some piecewise homeomorphisms of the circle, for surface groups given by some particular group presentations.In this talk I’ll try to explain the reverse the question: How to construct a group from a piecewise homeomorphism on the circle and verify it is indeed a surface group? This is a geometrization problem and I will describe a construction for a class of such piecewise homeomorphisms of the circle.