Let $C/\mathbb Q$ be a smooth projective curve of genus $g\geq 2$ or a smooth affine curve of genus $g \geq 1$. The exact sequence


$1 \to \pi\_1^{et}(C \otimes\_\mathbb Q \bar{\mathbb Q}) \to \pi\_1^{et}(C) \to \operatorname{Gal}(\bar{\mathbb Q}|\mathbb Q) \to 1$


gives a homomorphism from $\operatorname{Gal}(\bar{\mathbb Q}|\mathbb Q)$ to the outer automorphism group of $\pi\_1^{et}(C \otimes\_\mathbb Q \bar{\mathbb Q})$.


Is this homomorphism always injective?


If one instead takes $C$ to be a curve of genus $0$ with $3$ points removed, it is injective, by Belyi's Theorem.



Are all anabelian Galois actions faithful?
The answer is "yes" I think, even if you replace $\mathbb Q$ with a number field.


In the affine case this is a result of Matsumoto, as pointed out by Felipe Voloch, see


Matsumoto, Makoto 
Galois representations on profinite braid groups on curves. 
J. Reine Angew. Math. 474 (1996), 169–219.


In the proper case this is a more recent result of Hoshi and Mochizuki, see


Hoshi, Yuichiro; Mochizuki, Shinichi 
On the combinatorial anabelian geometry of nodally nondegenerate outer representations. 
Hiroshima Math. J. 41 (2011), no. 3, 275–342.


See also Tamás Szamuely's nice survey:


Heidelberg lectures on fundamental groups, in J. Stix (ed.) The Arithmetic of Fundamental Groups (PIA 2010), Contributions in Mathematical and Computational Sciences, Vol. 2, Springer-Verlag, 2012, 53--73


available here <http://www.renyi.hu/~szamuely/pia.pdf>


