Geodesic
Subgroups of One-Relator Fuchsian Groups
Canadian journal of mathematics, Tome 25 (1973) no. 4, pp. 888-891

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Let Γ be the Fuchsian group with canonical presentation:Generators:Relations:where . The genus g, the period partition {m1, m2 , ... , mr} and the number s are invariants of Γ. The symbol (g|m1, m2, ... , mr|s) is called the signature of Γ and it determines Γ up to isomorphism. 
Best, L. A. Subgroups of One-Relator Fuchsian Groups. Canadian journal of mathematics, Tome 25 (1973) no. 4, pp. 888-891. doi: 10.4153/CJM-1973-093-5
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[1] 1. Singerman, D., Subgroups of Fuchsian groups and finite permutation groups, Bull. London Math. Soc. 2 (1970), 319–323. Google Scholar

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