On parabolic and elliptic elements of the modular group

Authors: BİLAL DEMİR, ÖZDEN KORUOĞLU

Abstract: The modular group $\Gamma=PSL(2, \mathbf{Z})$ is isomorphic to the free product of two cyclic groups of orders $2$ and $3$. In this paper, we give a necessary and sufficient condition for the existence of elliptic and parabolic elements in $\Gamma$ with a given cusp point. Then we give an algorithm to obtain such elements in words of generators using continued fractions and paths in the Farey graph.

Keywords: Modular group, continued fractions, Farey graph

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