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Amenability, Poincaré series and quasiconformal maps

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Any 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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Authors and Affiliations

  1. Department of Mathematics, Princeton University, 08544, Princeton, NJ, USA

    Curt McMullen

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  1. Curt McMullen
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Research partially supported by an NSF Postdoctoral Fellowship

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McMullen, C. Amenability, Poincaré series and quasiconformal maps. Invent Math 97, 95–127 (1989). https://doi.org/10.1007/BF01850656

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