N-Rationality of Zeta Functions

The main result of the present paper is the sharpening of rationality w x w x w x theorems of Manning 23 , Fried 18 , and Gromov 19 , with an application to hyperbolic groups. The basic concept we shall use is that of Nrationality. A series is called N-rational if it is obtained from polynomials over N in Ž the variable t by applying the following operations: sum, product, star by definition the star of a series S is S* s Ý S, defined if S has no nG 0 . constant term: it is the inverse of 1 y S . The important condition is that one never performs subtraction; this may be interpreted by the existence of an algorithm which generates the objects whose generating function is the given series. Equivalently, a series is N-rational if and only it is the Ž . generating series of some rational regular language. Thus N-rationality is a kind of combinatorial rationality. An N-rational series is a rational series and has necessarily coefficients in N, but a rational series with coefficients in N need not be N-rational Ž w x see the book of Eilenberg 12, Example VIII.6.1 for a counterexample . w x and more on this subject . As a consequence of theorems of Berstel 3 , w x w x Soittola 27 , and Katayama et al. 20 , a complete characterization of Ž . N-rational series is known see Section 4.5 : roughly speaking, N-rational series are rational series with coefficients in N which have a unique pole of minimal modulus. Many rational series appearing in the mathematical literature are actually N-rational. This is the case for the Hilbert series of finitely generated