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Poincaré series and holomorphic averaging. (English) Zbl 0769.30036

We provide an alternate proof of McMullen’s theorem on contractive properties of the Poincaré series operator in the special case of universal covering. This case includes in particular Kra’s Theta conjecture. Our approach is to show that if the Poincaré series operators is insufficiently contractive then it is possible to average multiple-valued sections of affine bundles to obtain single-valued sections. But we show that such holomorphic averaging is in general impossible in the bundle of affine connections on the Riemann surface in question.

MSC:

30F99 Riemann surfaces
30F30 Differentials on Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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References:

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[5] [M] McMullen, C.: Amenability, Poincaré series and quasiconformal maps. Invent. Math.97, 95-127 (1989) · Zbl 0672.30017 · doi:10.1007/BF01850656
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