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Title: Commutator subgroups of the extended Hecke groups $\bar{H}(\lambda_q)$ (English)
Author: Sahin, R.
Author: Bizim, O.
Author: Cangul, I. N.
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 54
Issue: 1
Year: 2004
Pages: 253-259
Summary lang: English
Category: math
Summary: Hecke groups $H(\lambda _q)$ are the discrete subgroups of ${\mathrm PSL}(2,\mathbb{R})$ generated by $S(z)=-(z+\lambda _q)^{-1}$ and $T(z)=-\frac{1}{z} $. The commutator subgroup of $H$($\lambda _q)$, denoted by $H^{\prime }(\lambda _q)$, is studied in [2]. It was shown that $H^{\prime }(\lambda _q)$ is a free group of rank $q-1$. Here the extended Hecke groups $\bar{H}(\lambda _q)$, obtained by adjoining $R_1(z)=1/\bar{z}$ to the generators of $H(\lambda _q)$, are considered. The commutator subgroup of $\bar{H}(\lambda _q)$ is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the $H(\lambda _q)$ case, the index of $H^{\prime }(\lambda _q)$ is changed by $q$, in the case of $\bar{H}(\lambda _q)$, this number is either 4 for $q$ odd or 8 for $q$ even. (English)
Keyword: Hecke group
Keyword: extended Hecke group
Keyword: commutator subgroup
MSC: 11F06
MSC: 20H05
MSC: 20H10
idZBL: Zbl 1053.11038
idMR: MR2040237
Date available: 2009-09-24T11:11:59Z
Last updated: 2020-07-03
Stable URL:
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Reference: [3] H. S. M.  Coxeter and W. O. J. Moser: Generators and Relations for Discrete Groups.Springer, Berlin, 1957. MR 0088489
Reference: [4] E. Hecke: Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichungen.Math. Ann. 112 (1936), 664–699. MR 1513069, 10.1007/BF01565437
Reference: [5] D. L. Johnson: Topics in the Theory of Group Presentations. L.M.S. Lecture Note Series  42.Cambridge Univ. Press, Cambridge, 1980. MR 0695161
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