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Commutator subgroups of the extended Hecke groups H ¯ ( λ q )

Recep ŞahinOsman BizimI. N. Cangul — 2004

Czechoslovak Mathematical Journal

Hecke groups H ( λ q ) are the discrete subgroups of P S L ( 2 , ) generated by S ( z ) = - ( z + λ q ) - 1 and T ( z ) = - 1 z . The commutator subgroup of H ( λ q ) , denoted by H ' ( λ q ) , is studied in [2]. It was shown that H ' ( λ q ) is a free group of rank q - 1 . Here the extended Hecke groups H ¯ ( λ q ) , obtained by adjoining R 1 ( z ) = 1 / z ¯ to the generators of H ( λ q ) , are considered. The commutator subgroup of H ¯ ( λ q ) is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the H ( λ q ) case, the index of H ' ( λ q ) is changed by q , in the case of H ¯ ( λ q ) , this number is either 4 for...

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