Pro-$p$ Grothendieck Conjecture for Hyperbolic Polycurves

Abstract

In the present paper, we study the geometrically pro-$p$ fundamental groups of hyperbolic polycurves, i.e., successive extensions of families of hyperbolic curves. Among other results, we show that the isomorphism class of a hyperbolic polycurve of dimension $\le 4$ over a sub-$p$-adic field satisfying a certain group-theoretic condition is completely determined by the geometrically pro-$p$ fundamental group equipped with surjection onto the absolute Galois group of the base field.