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

TL;DR

This paper investigates the shortest possible representations of matrices in PSL(2,Z) using continuant matrices with integer coefficients, extending previous work that only considered positive coefficients, and explores conditions for uniqueness and conjugacy class generalizations.

Contribution

It introduces a method to find minimal integer coefficient presentations of PSL(2,Z) matrices using continuant matrices and analyzes their uniqueness and conjugacy class extensions.

Findings

Established criteria for minimal integer coefficient presentations.

Identified conditions for the uniqueness of such minimal presentations.

Generalized results to conjugacy classes in PSL(2,Z).

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 christian.streib@online.de.