On groups related to the Hecke groups

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.