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.


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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