On rationality of the cogrowth series

Kouksov, Dmitri

doi:10.1090/S0002-9939-98-04741-8

On rationality of the cogrowth series
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Proc. Amer. Math. Soc. 126 (1998), 2845-2847 Request permission

Abstract:

The cogrowth series of a group $G$ depends on the presentation of the group. We show that the cogrowth series of a non-empty presentation is a rational function not equal to 1 if and only if $G$ is finite. Except for the trivial group, this property is independent of presentation.

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