Amenability, Poincaré series and quasiconformal maps

SummaryAny coveringY→X of a hyperbolic Riemann surfaceX of finite area determines an inclusion of Teichmüller spaces Teich(X)↪Teich(Y). We show this map is an isometry for the Teichmüller metric if the covering isamenable, and contracting otherwise. In particular, we establish ‖Θ‖<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.

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