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

Hoshi, Y *(Kyoto)*

Thursday 27 August 2009, 14:00-15:00

Seminar Room 1, Newton Institute

#### Abstract

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.

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