Automatic structures and growth functions for finitely generated abelian groups

2011/12/08 by Satoshi Kamei, Kamei, Satoshi
Mathematics · #20F65 #68R15 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #math.GR #math.GT #msc:20F65 #msc:68R15

paper · pdf · doi:10.48550/arxiv.1112.1771

11 pages, 5 figures

arxiv created 2011/12/12 · arxiv updated 2011/12/13

Abstract

In this paper, we consider the formal power series whose n-th coefficient is the number of copies of a given finite graph in the ball of radius n centred at the identity element in the Cayley graph of a finitely generated group and call it the growth function. Epstein, Iano-Fletcher and Uri Zwick proved that the growth function is a rational function if the group has a geodesic automatic structure. We compute the growth function in the case where the group is abelian and see that the denominator of the rational function is determined from the rank of the group.

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