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On groups related to the Hecke groups. (English) Zbl 0782.11013

Seien \(A=({1\atop 0} {\lambda_ 1 \atop 1})\), \(B=({1\atop \lambda_ 2} {0\atop 1})\) mit \(\lambda_ 1,\lambda_ 2\in\mathbb{R}\); \(\lambda_ 1,\lambda_ 2>0\); betrachtet als zwei parabolische Elemente der \(\text{PSL}(2,\mathbb{R})\) und \(K(\lambda_ 1,\lambda_ 2)\) die von \(A\) und \(B\) erzeugte Gruppe. \(K(\lambda_ 1,\lambda_ 2)\) ist genau dann diskret, wenn \(\lambda_ 1\lambda_ 2\geq 4\) oder \(\lambda_ 1\lambda_ 2=4\cos^ 2(\pi/p)\), wobei \(p\in\mathbb{N}\), \(p\geq 3\).

MSC:

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

[1] S. Bochner, Some properties of modular relations, Ann. of Math. (2) 53 (1951), 332 – 363. · Zbl 0042.32101 · doi:10.2307/1969546
[2] K. Chandrasekharan and Raghavan Narasimhan, Hecke’s functional equation and arithmetical identities, Ann. of Math. (2) 74 (1961), 1 – 23. · Zbl 0107.03702 · doi:10.2307/1970304
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