Poincaré series and holomorphic averaging (Q1209122)

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.