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

TL;DR

This paper extends the classical connection between geodesics on modular surfaces and continued fractions to odd, grotesque, and even variants, revealing new relationships with specific subgroups of PSL(2,Z).

Contribution

It introduces a novel framework linking geodesics on certain modular surfaces to odd, grotesque, and even continued fractions, expanding the classical theory.

Findings

Established connection between geodesics on $ ext{PSL}(2, ext{Z})ackslash ext{H}$ and regular continued fractions.

Extended the connection to geodesics on $ ext{Gamma}ackslash ext{H}$ and odd, grotesque continued fractions.

Discussed the relationship between geodesics on $ ext{Theta}ackslash ext{H}$ and even continued fractions.

Abstract

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 $\left( \begin{smallmatrix} 1 & 2 \\ 0 & 1 \end{smallmatrix} \right)$.