Estimates For Simple Random Walks On Fundamental Groups Of Surfaces

Introduction Let \Gamma be a group generated by a finite set S which is symmetric (s 2 S () s \Gamma1 2 S) and which does not contain the unit element. Let X = Cay(\Gamma; S) be the Cayley graph with vertex set X 0 = \Gamma and, for x; y 2 \Gamma; with fx; yg an edge if x \Gamma1 y 2 S: The corresponding Markov operator MX is defined on functions on X 0 by (MX f) (x) = 1 k X y¸x f(y) f : \Gamma ! C x 2 \Gamma where k = jSj&lt