Geometry of Teichmüller space with the Teichmüller metric

The purpose of this chapter is to describe recent progress in the study of Teichmüller geometry. We focus entirely on the Teichmüller metric. A survey of the very important Weil-Petersson metric can be found in [W]. The study of the Teichmüller metric has different aspects. One major theme in the subject is to what extent Teichmüller space with the Teichmüller metric resembles a metric of negative curvature, and to what extent it resembles a metric of nonnegative curvature. This theme will occupy much of this survey and we will describe results in both directions. With somewhat the same theme we describe recent results about the Teichmüller geodesic flow on moduli space. Along somewhat different lines we describe some recent important work of K. Rafi that gives a combinatorial description of the Teichmüller metric. Another important subject is the study of the action of the action of the mapping class group on Teichmüller space. We will discuss some very important recent work of Eskin, Mirzakhani and co-authors on counting problems for the mapping class group.We will also describe some recent joint work with Benson Farb on the Teichmüller geometry of moduli space. There has also been a great deal of recent work on the related topics of the SL(2,R) action on spaces of quadratic differentials, Veech groups and Veech surfaces. These fall outside the scope of this article. We refer to the article of Hubert, Lanneau and Moeller in these proceedings for a discussion of these last subjects. For general references for Teichmüller theory, and quasi-conformal mappings I refer to the books of L.Ahlfors ([A]), J. Hubbard ([H]) and A. Papdoupolous ([P]). For a reference to the mapping class group I refer to the book of N.Ivanov ([I1] and the recent book of B.Farb and D.Margalit ([FMa]). For references to the theory of quadratic differentials there are the books of Strebel ([St]) and F.Gardiner ([G]). See the paper of L.Bers ([B1]) for a discussion ∗Author is supported in part by the NSF.