In their 2008 paper "Torelli theorem for curves over finite fields" Bogomolov, Korotiaev and Tschinkel mention in the beginning of Section 9 that absolute Galois groups of curves over $\mathbb{F}\_p^{alg}$ are free profinite on countably many generators. However, they do not give reference, and after talking to some colleagues I am at a loss with regard to the status of this statement. 


How does one derive this fact (if it is true)?



absolute Galois group of the function field of a curve over $\mathbb{F}_p^{alg}$
The theorem is true. It seems to have been proved independently by Florian Pop and by David Harbater. In Pop's paper, it is the corollary on p. 556.


MR1334484 (96k:14011)   

Pop, Florian   

Étale Galois covers of affine smooth curves. The geometric case of a conjecture of Shafarevich. On Abhyankar's conjecture.   

Invent. Math. 120 (1995), no. 3, 555–578.   

<http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002112868>


Harbater's paper is the following.


MR1352282 (97b:14035)   

Harbater, David   

Fundamental groups and embedding problems in characteristic p. (English summary)   
 
Recent developments in the inverse Galois problem (Seattle, WA, 1993), 353–369,   

Contemp. Math., 186, Amer. Math. Soc., Providence, RI, 1995. 


