Amenability, Poincaré Series and Quasiconformal Maps

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.