${\displaystyle SL(2,\mathbb{Z})}$, les tresses à trois brins, le tore modulaire et $Au{t}^{+}(F_2)$

The action of $SL(2,\mathbf{Z})$ on the integer torus and its quotient by central symmetry and Artin’s presentation of three strings braid group $B_3$, produces a presentation with parabolic generators \pmatrix{1& -1\cr 0& 1\cr} and \pmatrix{1& 0\cr 1& 1\cr}. This braided presentation describes the action of the derived group on Poincaré’s half plane and its quotient the modular torus, just as Nielsen’s theorem giving the group of direct automorphisms of the free group on two generators as semi-direct product, amalgamated on the index $2$ subgroup of the center of $B_3$, of inner automorphisms with $B_3$.