An estimate from above of spectral radii of random walks on surface groups

Using Gabber's Lemma, we get new estimates of the spectral radius of the simple random walk on the fundamental group of the orientable closed surface of genus $g$, $g\ge2$. In order to get better numerical estimates we base our method on Cannon's classification of the group elements by their cone types. The method may as well be applied to many other groups and graphs with finite numbers of cone types.