Earthquakes in the hyperbolic plane

Abstract:

Thurston’s earthquake theorem in the hyperbolic plane states that any orientation- preserving homeomorphism of the circle arises as the boundary homeomorphism of a so- called earthquake map. The goal of this thesis is to define and study these transformations and to give a detailed proof of the earthquake theorem, following Thurston’s paper. We also define relative hyperbolic structures on the hyperbolic plane, discuss their relation with homeomorphisms of the circle and we deduce from the earthquake theorem that any two such structures can be related by an earthquake map.