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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On two-generator discrete groups of Möbius transformations


Author: De Lin Tan
Journal: Proc. Amer. Math. Soc. 106 (1989), 763-770
MSC: Primary 20H05; Secondary 11F06
MathSciNet review: 969527
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Abstract: Assume that Möbius transformations $ f$ and $ g$ generate a discrete group. We obtain the following generalizations of Jørgensen's inequalities. If $ {\text{tr}}(fg{f^{ - 1}}{g^{ - 1}}) \ne 1$, then $ \left\vert {{\text{t}}{{\text{r}}^2}(f) - 2} \right\vert + \left\vert {{\text{tr}}(fg{f^{ - 1}}{g^{ - 1}}) - 1} \right\vert \geq 1$. If $ {\text{tr}}(fg{f^{ - 1}}{g^{ - 1}}) = 1$, then either $ {\text{t}}{{\text{r}}^2}(f) = 2{\text{ort}}{{\text{r}}^2}(f) = 1{\text{or}}\left\vert {{\text{t}}{{\text{r}}^2}(f) - 2} \right\vert\frac{1}{2}$ and $ \left\vert {{\text{t}}{{\text{r}}^2}(f) - 1} \right\vert > \frac{1}{2}$. If $ {\text{t}}{{\text{r}}^2}(f) \ne 1$, then $ \left\vert {{\text{t}}{{\text{r}}^2}(f) - 1} \right\vert + \left\vert {{\text{tr}}(fg{f^{ - 1}}{g^{ - 1}})} \right\vert \geq 1$. If $ {\text{t}}{{\text{r}}^2}(f) = 1$ then either $ {\text{tr}}(fg{f^{ - 1}}{g^{ - 1}}) = 0{\text{ortr}}(fg{f^{ - 1}}{g^{ - 1}}) = 1$ and $ \left\vert {{\text{tr}}(fg{f^{ - 1}}{g^{ - 1}}) - 1} \right\vert\frac{1}{2}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0969527-3
PII: S 0002-9939(1989)0969527-3
Article copyright: © Copyright 1989 American Mathematical Society