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## Studia Mathematica

1998 | 131 | 1 | 89-94
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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 $γ_n$.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
89-94
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-11-03
Twórcy
autor
• Institute of Mathematics, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Bibliografia
• [1] J. M. Cohen, Cogrowth and amenability of discrete groups, J. Funct. Anal. 48 (1982), 301-309.
• [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] 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.
• [6] R. Szwarc, A short proof of the Grigorchuk-Cohen cogrowth theorem, Proc. Amer. Math. Soc. 106 (1989), 663-665.
• [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.
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