The \( K(\pi,1)\)-property for smooth marked curves over finite fields (Q2832311)

In recent papers (for instance: [Doc. Math., J. DMV 12, 441--471 (2007; Zbl 1169.11048); J. Reine Angew. Math. 640, 203--235 (2010; Zbl 1193.14041)]) the second author studied the ``\(K(\pi, 1)\)-property for \(p\)'' (the definition which uses a Hochschild-Serre spectral sequence is given in \S 1.1 of the present paper) for arithmetic curves whose function fields has characteristic different from \(p\). It was shown that the Galois group of the maximal unramified outside \(S\) and completely \(T\)-split pro-\(p\)-extension of a global field of characteristic different from \(p\) is often of cohomological dimension at most two. In the present paper, the authors consider the case of a smooth marked curve \((X, T)\) over a finite field of characteristic \(p\). They prove that \((X, T)\) has \(K(\pi, 1)\)-property if \(X\) is affine, and give positive and negative examples in the case \(X\) is proper. They also consider the case of unmarked proper curves over a finite field of characteristic different from \(p\). The proofs use local and global computation of the étale cohomology groups.