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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 groups related to the Hecke groups


Authors: Marvin I. Knopp and Morris Newman
Journal: Proc. Amer. Math. Soc. 119 (1993), 77-80
MSC: Primary 20H10
MathSciNet review: 1152280
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Abstract: Let $ \left[ {\begin{array}{*{20}{c}} 1 & {{\lambda _1}} \\ 0 & 1 \\ \end{array} } \... ...[ {\begin{array}{*{20}{c}} 1 & 0 \\ {{\lambda _2}} & 1 \\ \end{array} } \right]$ be parabolic elements of $ \operatorname{PSL} (2,R)$, where $ {\lambda _1},{\lambda _2} > 0$. The principal result shown here is that $ K({\lambda _1},{\lambda _2})$, the group generated by these elements, is discrete if and only if $ {\lambda _1}{\lambda _2} \geqslant 4$, or $ {\lambda _1}{\lambda _2} = 4{\cos ^2}(\pi /p)$, where $ p$ is an integer $ \geqslant 3$. When $ {\lambda _1}{\lambda _2} = 4{\cos ^2}(\pi /p),\;K({\lambda _{1,}}{\lambda _2})$ is conjugate to the classical Hecke group $ H(2\cos (\pi /p))$ if $ p$ is odd; while if $ p$ is even, $ K({\lambda _1},{\lambda _2})$ is conjugate to a subgroup of $ H(2\cos (\pi /p))$ of index $ 2$. When $ {\lambda _1}{\lambda _2} \geqslant 4,\;K({\lambda _1},{\lambda _2})$ is conjugate to a subgroup of $ H(\sqrt {({\lambda _1}{\lambda _2})} )$ of index $ 2$. In all of these cases $ K({\lambda _1},{\lambda _2})$ is the free product of two cyclic groups.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1152280-1
PII: S 0002-9939(1993)1152280-1
Article copyright: © Copyright 1993 American Mathematical Society