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

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