The ratio and generating function of cogrowth coefficients of finitely generated groups

Szwarc, Ryszard

Szwarc, Ryszard

Studia Mathematica, Tome 129 (1998), p. 89-94 / Harvested from The Polish Digital Mathematics Library

Résumé

Let G be a group generated by r elements $g_{1}, \dots, g_{r}$ . Among the reduced words in $g_{1}, \dots, g_{r}$ of length n some, say $γ_{n}$ , represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of $γ_{2 n}$ has a limit, called the cogrowth exponent with respect to the generators $g_{1}, \dots, g_{r}$ . We show by analytic methods that the numbers $γ_{n}$ vary regularly, i.e. the ratio $γ_{2 n + 2} / γ_{2 n}$ is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function associated with the coefficients $γ_{n}$ .

Publié le : 1998-01-01

Zbl 0922.20039

EUDML-ID : urn:eudml:doc:216565

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     author = {Ryszard Szwarc},
     title = {The ratio and generating function of cogrowth coefficients of finitely generated groups},
     journal = {Studia Mathematica},
     volume = {129},
     year = {1998},
     pages = {89-94},
     zbl = {0922.20039},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv131i1p89bwm}
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Szwarc, Ryszard. The ratio and generating function of cogrowth coefficients of finitely generated groups. Studia Mathematica, Tome 129 (1998) pp. 89-94. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv131i1p89bwm/

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