  It is proved that the Green's function of the simple random walk on a surface
group of large genus decays exponentially at the spectral radius. It is also
shown that Ancona's inequalities extend to the spectral radius R, and therefore
that the Martin boundary for R-potentials coincides with the natural geometric
boundary S^1. This implies that the Green's function obeys a power law with
exponent 1/2 at the spectral radius.
