On groups related to the Hecke groups
HTML articles powered by AMS MathViewer
- by Marvin I. Knopp and Morris Newman PDF
- Proc. Amer. Math. Soc. 119 (1993), 77-80 Request permission
Abstract:
Let $\left [ {\begin {array}{*{20}{c}} 1 & {{\lambda _1}} \\ 0 & 1 \\ \end {array} } \right ],\left [ {\begin {array}{*{20}{c}} 1 & 0 \\ {{\lambda _2}} & 1 \\ \end {array} } \right ]$ 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.References
- S. Bochner, Some properties of modular relations, Ann. of Math. (2) 53 (1951), 332–363. MR 47719, DOI 10.2307/1969546
- K. Chandrasekharan and Raghavan Narasimhan, Hecke’s functional equation and arithmetical identities, Ann. of Math. (2) 74 (1961), 1–23. MR 171761, DOI 10.2307/1970304 M. I. Knopp, Results related to Hamburger’s theorem (in preparation).
- Carl Siegel, Bemerkung zu einem Satz von Hamburger über die Funktionalgleichung der Riemannschen Zetafunktion, Math. Ann. 86 (1922), no. 3-4, 276–279 (German). MR 1512091, DOI 10.1007/BF01457989
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 119 (1993), 77-80
- MSC: Primary 20H10
- DOI: https://doi.org/10.1090/S0002-9939-1993-1152280-1
- MathSciNet review: 1152280