Quasi-projections in Teichmüller space

Abstract

We consider a geometric property of the closest-points projection to a geodesic in Teichmüller space: the projection is called contracting if arbitrarily large balls away from the geodesic project to sets of bounded diameter. (This property always holds in negatively curved spaces.) It is shown here to hold if and only if the geodesic is precompact, i.e. its image in the moduli space is contained in a compact set. Some applications are given, e.g. to stability properties of certain quasi-geodesics in Teichmüller space, and to estimates of translation distance for pseudo-Anosov maps.