$SL(2,\mathbb{Z})$, les tresses \`a trois brins et le tore modulaire

TL;DR

This paper explores the action of the modular group SL(2,Z) on the integer torus, its quotient, and the braid group with three strands, analyzing their algebraic structures and geometric actions on the Poincaré half-plane.

Contribution

It provides a presentation of the three-strand braid group derived from the action of SL(2,Z) on the torus and describes its geometric action on the Poincaré half-plane.

Findings

Presentation of the braid group with parabolic generators

Description of the action on the Poincaré half-plane

Connection between SL(2,Z) and braid group structures

Abstract

The action of $SL(2,\mathbb{Z})$ on the integer torus and its quotient by central symmetry and Artin's presentation of three strings braids, produces a presentation with parabolic generators $\pmatrix{1& -1\cr 0& 1\cr}$ and $\pmatrix{1& 0\cr 1& 1\cr}$ and describes the action of the derived group on Poincar\'e's half plane.