Cohomology of the geometric fundamental group of hyperbolic polycurves

In the present paper, we study the cohomology groups of profinite groups obtained by successive extensions of families of (pro-Σ) surface groups. As an application, we show, among others, that the dimension of a hyperbolic polycurve (i.e., a successive extension of a family of hyperbolic curves) over a field of characteristic zero can be reconstructed group-theoretically from its geometric fundamental group.