Applications Of Teichmüller Theory In The Study Of Mapping Class Groups

As more aspects of geometric structures of the three dimensional manifolds have been known,people discovered that geometric structures can be understood as the result of gluing finitely many extremely simple homogeneous geometric pieces by mappings and the complexity of geometric structures of the manifold carried by the gluing mappings.Thus the mapping class groups is a very active research object.In all of those methods to study mapping class groups,studying mapping class groups through considering their actions on some special spaces are more fruitful than others.William.Thurston's proof of the classification of the elements of mapping class groups,the one that often regarded as a starting point towards current research,can be said to reflect this method perfectly.This thesis is a reading report.The main purpose of the thesis is to understand the classification theorem and its related tools and theories so that the author will have a solid foundation in this field after finishing this thesis.Thus,no important original results have been claimed by the author.The first chapter is a brief introduction to backgrounds of the Teichmiiller space and measured foliations.In this chapter,different with references,the author calculated the Teichmuller space of the torus in a more direct way.The second chapter mainly introduce some theorems on Moduli spaces,and the author gave a simple proof for Wolpert lemma by using covering space.The last chapter begin with a sketch of the proof of the Nielsen-Thurston classification theorem,and then discussed pseudo-Anosov mappings,which are the most important classes,through constructing some examples using methods from references.