Complete growth functions of hyperbolic groups

Abstract

. We study a generalization of the growth functions of finitely generated groups, namely the growth functions Σ _{g ∈ G} gz ^{| g |} with coefficients in the group ring ℤ[G]. Rationality and methods of computation of such functions are discussed, in particular for hyperbolic groups. The complete growth functions of surface groups are explicitly computed. The operator and geodesic growth functions are also studied.