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On the combinatorial cuspidalizations and the faithfulness of the outer Galois representations of hyperbolic curves

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Non-Abelian Fundamental Groups in Arithmetic Geometry

In this talk, I discuss the combinatorial anabelian geometry for nodally nondegenerate outer representations on the fundamental groups of hyperbolic curves. I plan to explain a result of a combinatorial version of the Grothendieck conjecture for nodally nondegenerate outer representations obtained in the joint work with Shinichi Mochizuki. As an application, we prove the injectivity portion of the combinatorial cuspidalization. We also generalize, by means of this injectivity result, the faithfulness proven by Makoto Matsumoto of the outer representation of the absolute Galois group on the profinite fundamental group of an affine hyperbolic curve over a certain field (e.g. number or p-adic local field) to the case where the given hyperbolic curve is proper.

This talk is part of the Isaac Newton Institute Seminar Series series.

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