# Amenability, Poincaré Series and Quasiconformal Maps

 Title: Amenability, Poincaré Series and Quasiconformal Maps Author: McMullen, Curtis T. Citation: McMullen, Curtis T. 1989. Amenability, Poincare series and quasiconformal maps. Inventiones Mathematicae 97(1): 95–127. Full Text & Related Files: McMullen_AmenabilityPoincare.pdf (3.113Mb; PDF) Abstract: Any covering $$Y \rightarrow X$$ of a hyperbolic Riemann surface$$X$$ of finite area determines an inclusion of Teichmüller spaces $$Teich(X) \hookrightarrow Teich(Y)$$. We show this map is an isometry for the Teichmüller metric if the covering isamenable, and contracting otherwise. In particular, we establish $$\|\Theta\|<1$$ for classical Poincaré series (Kra's "Theta conjecture"). The appendix develops the theory of geometric limits of quadratic differentials, used in this paper and a sequel. Published Version: doi:10.1007/BF01850656 Other Sources: http://www.math.harvard.edu/~ctm/papers/index.html Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:3446032 Downloads of this work: