On the combinatorial anabelian geometry of nodally nondegenerate outer representations

Hoshi, Yuichiro; Mochizuki, Shinichi

doi:10.32917/hmj/1323700038

Cited by 34 publications

(31 citation statements)

References 11 publications

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“…To conclude, we then have to show that instead the centralizer of ι inΠ g (2g + 2) C is trivial. By the short exact sequence (15) and induction on n, this last claim is implied by the lemma: Lemma 6.6. For all 0 ≤ n ≤ 2g + 1 (where, for g = 1, we let n ≥ 1), the subgroup ι is self-normalizing inΠ C g,n · ι .…”

Section: To See the Other Inclusion Kerˆ Cmentioning

confidence: 92%

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Congruence topologies on the mapping class group

Boggi

2020

Journal of Algebra

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Let Γ(S) be the pure mapping class group of a connected orientable surface S of negative Euler characteristic. For C a class of finite groups, letπ 1 (S) C be the pro-C completion of the fundamental group of S. The C -congruence completionΓ(S) C of Γ(S) is the profinite completion induced by the embedding Γ(S) → Out(π 1 (S) C ). In this paper, we begin a systematic study of such completions for different C . We show that the combinatorial structure of the profinite groupsΓ(S) C closely resemble that of Γ(S). A fundamental question is how C -congruence completions compare with pro-C completions. Even though, in general (e.g. for C the class of finite solvable groups),Γ(S) C is not even virtually a pro-C group, we show that, for Z/2 ∈ C , g(S) ≤ 2 and S open, there is a natural epimorphism from the C -congruence com-pletionΓ(S)(2) C of the abelian level of order 2 to its pro-C completion Γ(S)(2) C . In particular, this is an isomorphism for the class of finite groups and for the class of 2-groups. Moreover, in these two cases, the result also holds for a closed surface.

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Section: To See the Other Inclusion Kerˆ Cmentioning

confidence: 92%

“…Theorem 2.2 in [18]) and then extended by Hoshi and Mochizuki to the closed surface case (cf. Corollary 6.2 in[15] and Lemma 20 in[13]).…”

mentioning

confidence: 91%

Congruence topologies on the mapping class group

Boggi

2020

Journal of Algebra

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“…It follows from the constructions of α 0 , α ∆ 2 that γ Y • α 0 and [α ∆ 2 ] • γ X coincide after composing with the natural morphism Out FC (∆ Y 2 ) → Out(∆ Y ). On the other hand, since Out FC (∆ Y 2 ) → Out(∆ Y ) is injective (cf., e.g., [5], Theorem A), we conclude that γ Y • α 0 = [α ∆ 2 ] • γ X . Therefore, by applying the natural isomorphisms Π X 2 ∼ = ∆ X 2 out G K and Π Y 2 ∼ = ∆ Y 2 out G L , we obtain an isomorphism Π X 2 ∼ = Π Y 2 , which satisfies the required uniqueness and compatibility properties (cf.…”

Section: Cuspidalization Problems For Hyperbolic Curvesmentioning

confidence: 64%

“…Since the natural morphism Out FC (∆ (X× K K\{x}) n−1 ) → Out FC (∆ (X× K K\{x}) n−2 ) is injective (cf. [5]), we may carry out a similar argument to the above discussion by replacing G K by Π X and ∆ X 2 by ∆ (X× K K\{x}) n−1 . Hence, for j = 1, · · · , n, we obtain an isomorphism α j n : Π Xn for i = 1, · · · , n − 1.…”

Section: Cuspidalization Problems For Hyperbolic Curvesmentioning

confidence: 90%

On the cuspidalization problem for hyperbolic curves over finite fields

Wakabayashi¹

2016

Kyoto J. Math.

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Abstract. In this paper, we study some group-theoretic constructions associated to arithmetic fundamental groups of hyperbolic curves over finite fields. One of the main results of this paper asserts that any Frobeniuspreserving isomorphism between the geometrically pro-l fundamental groups of hyperbolic curves with one given point removed induces an isomorphism between the geometrically pro-l fundamental groups of the hyperbolic curves obtained by removing other points. Finally, we apply this result to obtain results concerning certain cuspidalization problems for fundamental groups of (not necessarily proper) hyperbolic curves over finite fields.

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“…A weaker version of Theorems 4.1 and 4.2 was established by the authors in a previous version of this paper that was available through the authors' websites. At the time we wrote that we were not aware of the article [23] by Hoshi and Mochizuki and so the only homomorphisms ζ lmn we knew to be injective were ζ ∞,∞,∞ and ζ 2,3,∞ , a result due to Belyi himself. That was already enough to prove faithfulness on the whole set of quasiplatonic curves but not on the more restricted subsets presented here.…”

Section: Faithfulness Of the Action Of Gal(q/q) On Regular Dessinsmentioning

confidence: 99%

The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces

González-Diez

Jaikin–Zapirain

2015

Proc. London Math. Soc.

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Beauville surfaces are an important kind of algebraic surfaces introduced by Catanese. They are rigid surfaces of general type defined over number fields. We prove that, for any σ∈Gal(double-struckQ¯/Q) different from the identity and the complex conjugation, there is a Beauville surface S such that S and its Galois conjugate Sσ have non‐isomorphic fundamental groups. This in turn easily implies that the action of Gal(double-struckQ¯/Q) on the set of isomorphism classes of Beauville surfaces is faithful. These results were conjectured by Bauer, Catanese and Grunewald, and immediately imply that Gal(double-struckQ¯/Q) acts faithfully on the connected components of the moduli space of surfaces of general type, a result due to the above‐mentioned authors. These results on Beauville surfaces heavily depend on the fact that the absolute Galois group acts faithfully on regular dessins, a result that we prove in this paper. Moreover, we show the stronger result that the action is faithful on the set of quasiplatonic (or triangle) curves of any given hyperbolic type.

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On the combinatorial anabelian geometry of nodally nondegenerate outer representations

Cited by 34 publications

References 11 publications

Congruence topologies on the mapping class group

Congruence topologies on the mapping class group

On the cuspidalization problem for hyperbolic curves over finite fields

The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces

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