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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
H (Xq) S C2 * so that the signature of H (Zq) is (O; 2, q, oo).
📌 3 Key Takeaways
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Objective & Hypothesis:Hecke groups H(Z) are the discrete subgroups of PSL(2, R) (the group of orientation preserving isometries of the upper half plane U) generated by two linear fractional transformations R (z) == - 1 / z and T (z) = z + X where XeR, X > 2 or X = Xq = 2cos (tt j q), qeN, q > 3^.
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Key Findings & Data:It is well-known that H (Xq) is the free product of two cyclic groups of orders 2 and q, i.e.
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Conclusion & Impact:H (Xq) S C2 * so that the signature of H (Zq) is (O; 2, q, oo).
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Normal subgroups · Hecke group · H (√2)
Abstract
Hecke groups H(Z) are the discrete subgroups of PSL(2, R) (the group of orientation preserving isometries of the upper half plane U) generated by two linear fractional transformations R (z) == - 1 / z and T (z) = z + X where XeR, X > 2 or X = Xq = 2cos (tt j q), qeN, q > 3^. These values of X are the only ones that give discrete groups, by a theorem of E. Hecke. We are going to be interested in the latter case /. = The element S = RT is then elliptic of order q. It is well-known that H (Xq) is the free product of two cyclic groups of orders 2 and q, i.e. H (Xq) S C2 * so that the signature of H (Zq) is (O; 2, q, oo).
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Cite This Article
Cangül, İ. (1994). Normal subgroups of the Hecke group H (√2). Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Volume: 43. https://doi.org/10.1501/Commua1_0000000479