  We consider the unit speed parametrization of the horocycle flow on infinite
Abelian covers of compact surfaces of negative curvature. We prove an
asymptotic result for the ergodic integrals of sufficiently regular functions.
In the case of constant curvature, where the unit speed and the uniformly
contracting parametrizations of horocycles coincide, we recover a result by
Ledrappier and Sarig. Our method, which does not use symbolic dynamics, is
based on a general Fourier decomposition for Abelian covers and on the study of
spectral theory of weighted (and twisted) transfer operators for the geodesic
flow acting on appropriate anisotropic Banach spaces. Finally, as a byproduct
result, we obtain a power deviation estimate for the horocycle ergodic averages
on compact surfaces, without requiring any pinching condition as in previous
results.
