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

Abstract

The connection between geodesics on the modular surface $\mathrm{PSL}(2,\mathbb{Z})\backslash \mathbb{H}$ 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 $\Gamma \cong {\mathbb{Z}}_3\ast {\mathbb{Z}}_3$ 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 $\Theta \backslash \mathbb{H}$ and even continued fractions is discussed in our framework, where $\Theta$ denotes the Theta subgroup of $\mathrm{PSL}(2,\mathbb{Z})$ generated by $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ and $\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right)$.

19 pages, minor typos corrected and clarifications added in the published version

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