The Grothendieck Conjecture for Hyperbolic Polycurves of Lower Dimension
Vol. 21 (2014), No. 2, Page 153–219.
Hoshi, Yuichiro
The Grothendieck Conjecture for Hyperbolic Polycurves of Lower Dimension
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Abstract:
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 $\textit{arbitrary dimension}$.
Keywords: Grothendieck conjecture, hyperbolic polycurve.
Mathematics Subject Classification (2010): Primary 14H30; Secondary 14H10, 14H25.
Mathematical Reviews Number: MR3288808
Received: 2012-11-08