$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Gal{Gal}\newcommand{\Z}{\mathbb{Z}}$In Grothendieck's *Esquisse* he claims that the action of
$$\Gal(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\widehat{\SL(2,\Z)})$$
obtained from the homotopy exact sequence of the étale fundamental group contains, after abelianization, all abelian $\ell$-adic representations defined by Jacobians and generalised Jacobians of algebraic curves defined over number fields. My questions (which might be hopelessly broad and misguided) are the following:

- How can one show this?
- Has any work been done extending this idea? A list of references is most welcome.

The original text is the following paragraph:

...from the point of view of Galois-Teichmüller theory, $\SL(2,\Z)$ can be considered as the fundamental “building block” of the "Teichmüller tower". The element of the structure of $\SL(2,\Z)$ which fascinates me above all is of course the outer action of $\Gal(\mathbb Q)$ on its profinite compactification. By Bielyi’s theorem, taking the profinite compactifications of subgroups of finite index of $\SL(2,\Z)$, and the induced outer action (up to also passing to an open subgroup of $\Gal(\mathbb Q)$),

we essentially find the fundamental groups of all algebraic curves(not necessarily compact)defined over number fields$K$, and the outer action of $\Gal(K)$ on them – at least it is true that every such fundamental group appears as a quotient of one of the first groups (*). Taking the “anabelian yoga” (which remains conjectural) into account, which says that an anabelian algebraic curve over a number field $K$ (finite extension of $\mathbb Q$) is known up to isomorphism when we know its mixed fundamental group (or what comes to the same thing, the outer action of $\Gal(K)$ on its profinite geometric fundamental group), we can thus say thatall algebraic curves defined over number fields are “contained” in the profinite compactification $\widehat{\SL(2, \Z)}$, and in the knowledge of a certain subgroup $\Gal(\mathbb Q)$ of its group of outer automorphisms! Passing to the abelianisations of the preceding fundamental groups, we see in particular that all the abelian $\ell$-adic representations dear to Tate and his circle, defined by Jacobians and generalised Jacobians of algebraic curves defined over number fields, are contained in this single action of $\Gal(\mathbb Q)$ on the anabelian profinite group $\widehat{\SL(2, \Z)}$!