TITLE: Group cohomology of fundamental group of a curve
QUESTION [8 upvotes]: The question should be an elementary result in the theory of etale cohomology, but I failed to understand it because I am a complete beginner of the theory. So, I should apologise in advance for this rudimenatry question.
Let $C$ be a connected hyperbolic curve defined over an algebraically closed field of characteristic 0. Let us write $\pi_1$ for the maximal pro-$p$ quotient of the etale fundamental group of $C$ with some choice of base-point. There is a statement which claims that, for any integers $i,r$, there exists a natural morphism
$H^i(\pi_1,\mathbb{Z}_p(r))\to H^i(C,\mathbb{Z}_p(r))$
which in fact an isomoprhism.
My questions are:

How is the natural map defined?
Why is it an isomorphism?

Surely any comment would helpful, but if anyone can explain it with details, that would be greatly helpful.

REPLY [2 votes]: The best thing you could do in my opinion is to have a look at 
Appendix A  
Algebraic $K(\pi, 1)$ Spaces 
in
Stix, Jakob
Projective anabelian curves in positive characteristic and descent theory for log-étale covers.
Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 2002. 
http://www.ams.org/mathscinet-getitem?mr=2012864
http://www.math.uni-frankfurt.de/~stix/research/preprints/STIXdissB.pdf
where this issue (for the whole fundamental group and for the pro-$\ell$ quotient) is treated with great care.