Growth functions on Fuchsian groups and the Euler characteristic

Let G be a finitely generated group, and 2; a finite generating set for G. Then 2; determines a norm on G, called the word norm, defined by letting Ig[ be the minimal length ofg as a word in 27. Following Milnor [,15], we define the growth series g(z) = 1 + a2z +. . . + a,z"..., where a, is the number of elements of G of word norm exactly n. Much of the early work (e.g., [-1, 16, 17, 23]) on growth series was concerned with asymptotic properties of the coefficients of g(z). This work culminated in Gromov 's theorem [12] that G has polynomial growth if and only if it is virtually nilpotent and in Grigorchuk's theorem [11] that there are finitely generated groups whose growth is neither polynomial nor exponential. If G is a Coxeter group and 2; is the standard generating set for G, Bourbaki [3] showed that the growth series g(z) is the series of a rational function f (z) and Serre [-20] showed that f(1) = I/z(G), where z(G) is the rational Euler characteristic of G. Note that if G is finite then g(1)=order(G)=l/z(G), but g(z) is not defined at one if G is infinite. If G is a compact hyperbolic or irreducible Euclidean Coxeter group and 27 is the standard generating set for G, Serre showed that f (1 /z )= +_ f(z), and so the poles of f are algebraic units. In contrast to the work on the asymptotic properties of the coefficients of g(z), these results raised the possibility of a beautiful theory of the exact structure of growth functions on groups. However, a decade went by before there was much further work on this subject. In two papers ([5] and [6]) Cannon studied growth functions for cocompact hyperbolic groups. If G is a cocompact hyperbolic group and X is any finite generating set for G, he showed that the coefficients a, of the growth series g(z) satisfy a linear recursion, and hence g(z) is the power series of a rational function f(z). He also computed some examples when G is a closed surface group or a compact hyperbolic Coxeter group. If G is a closed surface group and 2; is a geometric generating set (see below for a definition) for G coming from a one relator presentation, then f is reciprocal ( f ( z )=f (1 /z ) ) , f (1)=l /x (G) , the