Complete growth functions of hyperbolic groups
Given a finitely generated group \(G\) and a finite system of generators \(S\) in \(G\), we study the formal power series \(F_{G,S}(z)=\sum_{g\in G}gz^{|g|}\) with coefficients in the group ring \(\mathbb{Z}[G]\), which we call the complete growth function of \(G\) with respect to \(S\). We discuss the problem of rationality of such power series, in particular for hyperbolic groups, and methods of computations. The complete growth functions of surface groups are explicitly computed. The coefficients of this power series may also be viewed as elements of the reduced \(C^*\)-algebra of the group. We establish a relation between the radius of convergence of the operator growth series and the growth rate of the group. The notion of complete growth series has sense for any formal language over a finite alphabet. An interesting example provided by groups is the complete growth series of the language of geodesics of \(G\) with respect to \(S\). We discuss its rationality and compute some examples.
scientific article; zbMATH DE number 1058850