The Grothendieck Conjecture for Hyperbolic Polycurves of Lower Dimension

In the present paper, we discuss Grothendieck’s anabelian conjecture for hyperbolic polycurves, i.e., successive extensions of families of hyperbolic curves. One of the consequences obtained in the present paper is that the isomorphism class of a hyperbolic polycurve of dimension less than or equal to four over a sub-p-adic field is completely determined by its étale fundamental group (i.e., which we regard as being equipped with the natural outer surjection of the étale fundamental group onto a fixed copy of the absolute Galois group of the base field). We also verify the finiteness of certain sets of outer isomorphisms between the étale fundamental groups of hyperbolic polycurves of arbitrary dimension