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

Studia Mathematica (1998)

- Volume: 131, Issue: 1, page 89-94
- ISSN: 0039-3223

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topSzwarc, Ryszard. "The ratio and generating function of cogrowth coefficients of finitely generated groups." Studia Mathematica 131.1 (1998): 89-94. <http://eudml.org/doc/216565>.

@article{Szwarc1998,

abstract = {Let G be a group generated by r elements $g_1,…,g_r$. Among the reduced words in $g_1,…,g_r$ 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 $γ_\{2n\}$ has a limit, called the cogrowth exponent with respect to the generators $g_1,…,g_r$. We show by analytic methods that the numbers $γ_n$ vary regularly, i.e. the ratio $γ_\{2n+2\}/γ_\{2n\}$ is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function associated with the coefficients $γ_n$.},

author = {Szwarc, Ryszard},

journal = {Studia Mathematica},

keywords = {cogrowth of subgroups; free group; amenable groups; free groups; finitely generated groups; generating functions},

language = {eng},

number = {1},

pages = {89-94},

title = {The ratio and generating function of cogrowth coefficients of finitely generated groups},

url = {http://eudml.org/doc/216565},

volume = {131},

year = {1998},

}

TY - JOUR

AU - Szwarc, Ryszard

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

JO - Studia Mathematica

PY - 1998

VL - 131

IS - 1

SP - 89

EP - 94

AB - Let G be a group generated by r elements $g_1,…,g_r$. Among the reduced words in $g_1,…,g_r$ 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 $γ_{2n}$ has a limit, called the cogrowth exponent with respect to the generators $g_1,…,g_r$. We show by analytic methods that the numbers $γ_n$ vary regularly, i.e. the ratio $γ_{2n+2}/γ_{2n}$ is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function associated with the coefficients $γ_n$.

LA - eng

KW - cogrowth of subgroups; free group; amenable groups; free groups; finitely generated groups; generating functions

UR - http://eudml.org/doc/216565

ER -

## References

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- [3] H. Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 149-156.
- [4] G. Szegő, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ. 23, Providence, R.I., 1975.
- [5] R. Szwarc, An analytic series of irreducible representations of the free group, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 1, 87-110. Zbl0634.22003
- [6] R. Szwarc, A short proof of the Grigorchuk-Cohen cogrowth theorem, Proc. Amer. Math. Soc. 106 (1989), 663-665. Zbl0681.43004
- [7] S. Wagon, Elementary problem E 3226, Amer. Math. Monthly 94 (1987), 786-787.
- [8] W. Woess, Cogrowth of groups and simple random walks, Arch. Math. (Basel) 41 (1983), 363-370. Zbl0522.20043

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