Geodesic
Commutator subgroups of the extended Hecke groups $\bar{H}(\lambda_q)$
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 1, pp. 253-259 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 11F06, 20H05, 20H10
Keywords: Hecke group; extended Hecke group; commutator subgroup 
@article{CMJ_2004_54_1_a22,
     author = {Sahin, R. and Bizim, O. and Cangul, I. N.},
     title = {Commutator subgroups of the extended {Hecke} groups $\bar{H}(\lambda_q)$},
     journal = {Czechoslovak Mathematical Journal},
     pages = {253--259},
     year = {2004},
     volume = {54},
     number = {1},
     mrnumber = {2040237},
     zbl = {1053.11038},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2004_54_1_a22/}
}
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Sahin, R.; Bizim, O.; Cangul, I. N. Commutator subgroups of the extended Hecke groups $\bar{H}(\lambda_q)$. Czechoslovak Mathematical Journal, Tome 54 (2004) no. 1, pp. 253-259. http://geodesic.mathdoc.fr/item/CMJ_2004_54_1_a22/