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In this paper, we consider the formal power series whose n-th coefficient isthe number of copies of a given finite graph in the ball of radius n centred atthe identity element in the Cayley graph of a finitely generated group and callit the growth function. Epstein, Iano-Fletcher and Uri Zwick proved that thegrowth function is a rational function if the group has a geodesic automaticstructure. We compute the growth function in the case where the group isabelian and see that the denominator of the rational function is determinedfrom the rank of the group.

Automatic structures and growth functions for finitely generated abelian groups