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Keywords:
Hecke group; extended Hecke group; commutator subgroup
Hecke group; extended Hecke group; commutator subgroup
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.
References:
[1] R. B. J. T. Allenby: Rings, Fields and Groups. Second Edition. Edward Arnold, London-New York-Melbourne-Auckland, 1991. MR 1144518
[2] I. N. Cangül and D. Singerman: Normal subgroups of Hecke groups and regular maps. Math. Proc. Camb. Phil. Soc. 123 (1998), 59–74. DOI 10.1017/S0305004197002004 | MR 1474865
[3] H. S. M. Coxeter and W. O. J. Moser: Generators and Relations for Discrete Groups. Springer, Berlin, 1957. MR 0088489
[4] E. Hecke: Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichungen. Math. Ann. 112 (1936), 664–699. DOI 10.1007/BF01565437 | MR 1513069
[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
[6] G. A. Jones and J. S. Thornton: Automorphisms and congruence subgroups of the extended modular group. J. London Math. Soc. 34 (1986), 26–40. DOI 10.1112/jlms/s2-34.1.26 | MR 0859146
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