Affine groups of flat surfaces

Abstract

According to Thurston’s classification, a diffeomorphism of a closed oriented surface is either elliptic, reducible or a pseudo-Anosov diffeomorphism. The structure of pseudo-Anosov diffeomorphisms, e.g., their dilatation coefficients and the corresponding measured foliations have been intensely investigated since. Rather than focusing on properties of a single diffeomorphism, the purpose of this chapter is to study the flat surfaces the pseudo-Anosov diffeomorphisms live on together with their whole group of affine diffeomorphisms.