## The Grothendieck Conjecture for Hyperbolic Polycurves of Lower Dimension

J. Math. Sci. Univ. Tokyo
Vol. 21 (2014), No. 2, Page 153–219.

Hoshi, Yuichiro
The Grothendieck Conjecture for Hyperbolic Polycurves of Lower Dimension
In the present paper, we discuss Grothendieck's anabelian conjecture for $\textit{hyperbolic polycurves}$, 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 $\textit{completely determined}$ by its étale fundamental group (i.e., which we regard as being equipped with the natural outer surjection of the \'etale fundamental group onto a $\it{fixed}$ copy of the absolute Galois group of the base field). We also verify the $\it{finiteness}$ of certain sets of outer isomorphisms between the étale fundamental groups of hyperbolic polycurves of $\textit{arbitrary dimension}$.