Contraction properties of the Poincaré series operator.pdf

CONTRACTION PROPERTIES OFTHE POINCARE SERIES OPERATORDavid E. Barrett and Jeffrey DillerUniversity of MichiganCornell University1. IntroductionIn a striking pair of papers, McMullen gave a new proof of the contraction prop-erties of Thurstons \skinning map|an iteration on the Teichmuller space of aRiemann surface. His approach was to reduce the problem to the study of a push-forward operator (called the Poincare series operator) for quadratic dierentials[Mc2], and then show that this pushforward operator is itself contracting [Mc1].Our aim in this paper is to give new proofs of McMullens estimates on the normof the Poincare series operator. Our methods dier signicantly from McMullens,especially in that we avoid the notion of \amenability, and some of the relatedcombinatorial arguments, in favor of more complex analytic and geometric tactics.Our methods have the advantage of yielding estimates that are completely explicitin terms of the injectivity radii of the Riemann surfaces involved. On the otherhand, our methods address only the case of covering surfaces with nitely gener-ated fundamental group. This is not too serious a shortcoming, since McMullenuses only the nite topology case in his applications to the skinning mapIn the rest of this introduction, we will provide some basic denitions, stateour main results, and explain the organization of this paper. The introductionsto McMullens papers do a wonderful job of summarizing the connections betweenquadratic dierentials and Teichmuller theory, and between Teichmuller theory andThurstons program. A good reference on quadratic dierentials is [Ga]. Busersbook [Bu] oers a point of view on Riemann surfaces that is particularly well{suitedto the methods we use here.Let X be a Riemann surface. A quadratic dierential on X is an expressionof the form  = (z) dz2 in local coordinates. Put more abstractly, a quadraticdierential is a section of the square of the holomorphic cotangent bundle of X.  iscalled h