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Commutator subgroups of the extended Hecke groups \(\overline H (\lambda _q)\). (English) Zbl 1053.11038

Summary: Hecke groups \(H(\lambda _q)\) are the discrete subgroups of PSL\((2, \mathbb R)\) generated by \(S (z) = - (z + \lambda _q)^{-1}\) and \(T (z) = -1/z \). The commutator subgroup of \(H (\lambda _q)\), denoted by \(H' (\lambda _q)\), is studied. It was shown that \(H' (\lambda _q)\) is a free group of rank \(q - 1\). Here the extended Hecke groups \(\overline H (\lambda _q)\), obtained by adjoining \(R_1 (z) = 1/\overline z\) to the generators of \(H (\lambda _q)\), are considered. The commutator subgroup of \(\overline H (\lambda _q)\) 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 \(H' (\lambda _q)\) is changed by  \(q\), in the case of \(\overline H (\lambda _q)\), this number is either 4 for \(q\) odd or 8 for \(q\) even.

MSC:

11F06 Structure of modular groups and generalizations; arithmetic groups
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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References:

[1] R. B. J. T. Allenby: Rings, Fields and Groups. Second Edition. Edward Arnold, London-New York-Melbourne-Auckland, 1991. · Zbl 0726.00001
[2] I. N. Cangül and D. Singerman: Normal subgroups of Hecke groups and regular maps. Math. Proc. Camb. Phil. Soc. 123 (1998), 59-74. · Zbl 0893.20036
[3] H. S. M. Coxeter and W. O. J. Moser: Generators and Relations for Discrete Groups. Springer, Berlin, 1957. · Zbl 0077.02801
[4] E. Hecke: Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichungen. Math. Ann. 112 (1936), 664-699. · Zbl 0014.01601
[5] D. L. Johnson: Topics in the Theory of Group Presentations. L.M.S. Lecture Note Series 42. Cambridge Univ. Press, Cambridge, 1980.
[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. · Zbl 0576.20031
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