Minimal Presentation of $PSL(2,\mathbb{Z})$ Using Continuant Matrices with Integer Coefficients

Abstract

In this article, the goal is to find the shortest presentation of a matrix $A \in PSL(2,\mathbb{Z})$ in terms of the so-called continuant matrices which are most known for their role in continued fraction theory. In chapter 7 of arXiv:1811.01229, Morier-G\'enoud and Ovsienko investigate this problem with the restriction that all coefficients of the continuant matrices are positive. Now, the goal is to determine the shortest presentation allowing all integer coefficients. To determine this minimal presentation, a few characteristic transformations will be introduced. It will also be investigated under which conditions such a minimal presentation becomes unique. The results are also generalized on conjugacy classes in $PSL(2,\mathbb{Z})$. If you wish to contact the author or you have some questions related to this work, feel free to write an email to [e-mail].