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

Recep Şahin, Osman Bizim, I. 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...

Computing fundamental domains for Fuchsian groups

John Voight (2009)

Journal de Théorie des Nombres de Bordeaux

We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group Γ with cofinite area. As a consequence, we compute the invariants of Γ , including an explicit finite presentation for Γ .

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