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

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 Poincaré’s half plane.