Coding of geodesics on some modular surfaces and applications to odd and even continued fractions

The connection between geodesics on the modular surface $\operatorname{PSL}(2,{\mathbb Z})\backslash {\mathbb H}$ and regular continued fractions, established by Series, is extended to a connection between geodesics on $\Gamma\backslash {\mathbb H}$ and odd and grotesque continued fractions, where $\Gamma\cong {\Bbb Z}_3 \ast {\Bbb Z}_3$ is the index two subgroup of $\operatorname{PSL}(2,{\mathbb Z})$ generated by the order three elements $\left( \begin{smallmatrix} 0 & -1 \\ 1 & 1 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 0 & 1 \\ -1 & 1 \end{smallmatrix} \right)$, having an ideal quadrilateral as fundamental domain. A similar connection between geodesics on $\Theta\backslash {\mathbb H}$ and even continued fractions is discussed in our framework, where $\Theta$ denotes the Theta subgroup of $\operatorname{PSL}(2,{\mathbb Z})$ generated by $\left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 1 & 2 \\ 0 & 1 \end{smallmatrix} \right)$.