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

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.