$\displaystyle SL(2, {\Bbb Z})$, les tresses \`a trois brins, le tore modulaire et $Aut^{+}(F_{2})$

TL;DR

This paper explores the connections between the modular group, three-strand braids, and automorphisms of a free group, providing new insights into their algebraic and geometric structures.

Contribution

It establishes a novel presentation linking $SL(2, Z)$, braid groups, and automorphisms of free groups, enriching the understanding of their interrelations.

Findings

Presentation of $SL(2, Z)$ via braid generators

Description of the action on the modular torus

Identification of automorphism groups as semi-direct products

Abstract

The action of $SL(2, {\bf Z})$ 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 Poincar\'e'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 $B_{3}$.