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September 2010 On the combinatorial cuspidalization of hyperbolic curves
Shinichi Mochizuki
Osaka J. Math. 47(3): 651-715 (September 2010).

Abstract

In this paper, we continue our study of the pro-$\Sigma$ fundamental groups of configuration spaces associated to a hyperbolic curve, where $\Sigma$ is either the set of all prime numbers or a set consisting of a single prime number, begun in an earlier paper. Our main result may be regarded either as a combinatorial, partially bijective generalization of an injectivity theorem due to Matsumoto or as a generalization to arbitrary hyperbolic curves of injectivity and bijectivity results for genus zero curves due to Nakamura and Harbater--Schneps. More precisely, we show that if one restricts one's attention to outer automorphisms of such a pro-$\Sigma$ fundamental group of the configuration space associated to a(n) affine (respectively, proper) hyperbolic curve which are compatible with certain ``fiber subgroups'' (i.e., groups that arise as kernels of the various natural projections of a configuration space to lower-dimensional configuration spaces) as well as with certain cuspidal inertia subgroups, then, as one lowers the dimension of the configuration space under consideration from $n+1$ to $n \ge 1$ (respectively, $n \ge 2$), there is a natural injection between the resulting groups of such outer automorphisms, which is a bijection if $n \ge 4$. The key tool in the proof is a combinatorial version of the Grothendieck conjecture proven in an earlier paper by the author, which we apply to construct certain canonical sections.

Citation

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Shinichi Mochizuki. "On the combinatorial cuspidalization of hyperbolic curves." Osaka J. Math. 47 (3) 651 - 715, September 2010.

Information

Published: September 2010
First available in Project Euclid: 24 September 2010

zbMATH: 1207.14032
MathSciNet: MR2768498

Subjects:
Primary: 14H30
Secondary: 14H10

Rights: Copyright © 2010 Osaka University and Osaka City University, Departments of Mathematics

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Vol.47 • No. 3 • September 2010
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