# 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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## Abstract

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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 ${\gamma }_{n}$, represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of ${\gamma }_{2n}$ 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 ${\gamma }_{n}$ vary regularly, i.e. the ratio ${\gamma }_{2n+2}/{\gamma }_{2n}$ is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function associated with the coefficients ${\gamma }_{n}$.

## How to cite

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Szwarc, 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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1. [1] J. M. Cohen, Cogrowth and amenability of discrete groups, J. Funct. Anal. 48 (1982), 301-309. Zbl0499.20023
2. [2] R. I. Grigorchuk, Symmetrical random walks on discrete groups, in: Multicomponent Random Systems, R. L. Dobrushin and Ya. G. Sinai (eds.), Nauka, Moscow, 1978, 132-152 (in Russian); English transl.: Adv. Probab. Related Topics 6, Marcel Dekker, 1980, 285-325.
3. [3] H. Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 149-156.
4. [4] G. Szegő, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ. 23, Providence, R.I., 1975.
5. [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. [6] R. Szwarc, A short proof of the Grigorchuk-Cohen cogrowth theorem, Proc. Amer. Math. Soc. 106 (1989), 663-665. Zbl0681.43004
7. [7] S. Wagon, Elementary problem E 3226, Amer. Math. Monthly 94 (1987), 786-787.
8. [8] W. Woess, Cogrowth of groups and simple random walks, Arch. Math. (Basel) 41 (1983), 363-370. Zbl0522.20043

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