Geometriae Dedicata 53: 1-23, 1994. ] ~) 1994 Kluwer Academic Publishers. Printed in the Netherlands. Growth Functions for Semi-Regular Tilings of the Hyperbolic Plane* W. J. FLOYD 1 and S. P. PLOTNICK 2 1Department of Mathematics, VPI&SU, Blacksburg, VA 24061-0123, U.S.A. 2Department of Mathematics, SUNY at Albany, Albany, NY 12222, U.S.A. (Received: 2 November 1992; revised version: 25 January 1994) Abstract. This paper studies the growth function, with respect to the generating set of edge identi- fications, of a surface group with fundamental domain D in the hyperbolic plane an n-gon whose angles alternate between rr/p and rr/q. The possibilities of n, p and q for which a torsion-freesurface group can have such a fundamental polygon are classified, and the growth functions are computed. Conditions are given for which the denominator of the growth function is a product of cyelotomic polynomials and a Salem polynomial. Mathematics Subject Classifications (1991): Primary 20F32; secondary 22E40, 20F05, 57N05. 1. Introduction Let G be a finitely generated group, and let ~ be a finite generating set for G. Then the word norm l l = I lr. is defined on G by I1GI = 0andif 9 C Gand9 ¢ la then Ig[ = min{n: 9 = gl... gn and for each i C {1,..., n} either 9i E ~ or 9~ -1 E £}. The growth series for G is g(z) = ~=oanz ~, where a~ is the number of elements of G with word norm n. In [1], Cannon proved that if G is a discrete, cocompact group of isometries of hyperbolic space and ~ is any fnite generating set for G, then the growth series 9(z) is the power series of a rational function f(z), which we call the growth function. If G is a discrete, cocompact group of isometries of 1~ 2 and D is a fundamental polygon for the action of G, then the geometric generating set for (G, D) is ~ = {g C GIgD f) D is an edge of D}. Cannon and Wagreich ([2], [3], [6]) showed that if G is the fundamental group of a closed orientable surface of genus g > 2, and Y;. is the geometric generating set for a fundamental domain for G that is a 4g-gon, then the growth function 1 + 2z + 2z 2 + ... + 2z 2g-2 + 2z 2g-1 + z 2g = 1 q- (2- 4g)z+ (2- 4g)z 2 +... +(2 - 4g)z 2g-2 + (2 - 4g)z 2g-1 + z 2g * This work was supported in part by NSF Research Grants.