On the procongruence completion of the Teichmüller modular group

Boggi, Marco

doi:10.1090/s0002-9947-2014-05971-0

Cited by 9 publications

(75 citation statements)

References 18 publications

(72 reference statements)

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“…, s, and we denote by [γ] a cycle in H 1 (S K , Q) supported onγ and by , S K the standard symplectic bilinear form on H 1 (S K , Q). In Lemma 5.11 of [6], we showed that, when log(ρ U (τ mh 1 γ 1 · . .…”

Section: Centralizers Of Pro-c Multitwistsmentioning

confidence: 97%

“…It is not difficult to show (cf. Section 4 [6]) that these orbits correspond to the possible topological types of the surface S σ, for σ a multicurve on S.…”

Section: The Complex Of Pro-c Curves On Smentioning

confidence: 98%

“…The following theorem is a generalization to the pro-C case of Lemma 4.6 of [6]. Let us recall that a simple closed curve (an embedded circle) on S is peripheral if it bounds on one side a disc with a single puncture.…”

Section: The Closure Of Stabilizersmentioning

confidence: 99%

“…There is a natural simplicial action of Γ(S) on C(S). In this subsection, we will define a pro-C version of the complex of curves C(S), generalizing definitions and results given in Section 3 and 4 of [6].…”

Section: The Complex Of Pro-c Curves On Smentioning

confidence: 99%

“…Proof. In order to prove Theorem 3.5, we just need to rephrase the proof of Theorem 4.2 in [6] in terms of pro-C completions and C -congruence completions instead of full profinite completions and full congruence completions of all the objects involved, where Lemma 4.3 in [6] is replaced by its pro-C version (cf. Lemma 2.3 in this paper).…”

Section: The Complex Of Pro-c Curves On Smentioning

confidence: 99%

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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: Centralizers Of Pro-c Multitwistsmentioning

confidence: 97%

“…It is not difficult to show (cf. Section 4 [6]) that these orbits correspond to the possible topological types of the surface S σ, for σ a multicurve on S.…”

Section: The Complex Of Pro-c Curves On Smentioning

confidence: 98%

Section: The Closure Of Stabilizersmentioning

confidence: 99%

Section: The Complex Of Pro-c Curves On Smentioning

confidence: 99%

Section: The Complex Of Pro-c Curves On Smentioning

confidence: 99%

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Congruence topologies on the mapping class group

Boggi

2020

Journal of Algebra

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Topics Surrounding the Combinatorial Anabelian Geometry of Hyperbolic Curves Iv: Discreteness and Sections

HOSHI,

MOCHIZUKI

2024

Nagoya Math. J.

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Let $\Sigma $ be a nonempty subset of the set of prime numbers which is either equal to the entire set of prime numbers or of cardinality one. In the present paper, we continue our study of the pro- $\Sigma $ fundamental groups of hyperbolic curves and their associated configuration spaces over algebraically closed fields in which the primes of $\Sigma $ are invertible. The present paper focuses on the topic of comparison between the theory developed in earlier papers concerning pro- $\Sigma $ fundamental groups and various discrete versions of this theory. We begin by developing a theory concerning certain combinatorial analogues of the section conjecture and Grothendieck conjecture. This portion of the theory is purely combinatorial and essentially follows from a result concerning the existence of fixed points of actions of finite groups on finite graphs (satisfying certain conditions). We then examine various applications of this purely combinatorial theory to scheme theory. Next, we verify various results in the theory of discrete fundamental groups of hyperbolic topological surfaces to the effect that various properties of (discrete) subgroups of such groups hold if and only if analogous properties hold for the closures of these subgroups in the profinite completions of the discrete fundamental groups under consideration. These results make possible a fairly straightforward translation, into discrete versions, of pro- $\Sigma $ results obtained in previous papers by the authors. Finally, we discuss a construction that was considered previously by M. Boggi in the discrete case from the point of view of the present paper.

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Notes on hyperelliptic mapping class groups

Boggi

2023

Glasgow Math. J.

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Hyperelliptic mapping class groups are defined either as the centralizers of hyperelliptic involutions inside mapping class groups of oriented surfaces of finite type or as the inverse images of these centralizers by the natural epimorphisms between mapping class groups of surfaces with marked points. We study these groups in a systematic way. An application of this theory is a counterexample to the genus $2$ case of a conjecture by Putman and Wieland on virtual linear representations of mapping class groups. In the last section, we study profinite completions of hyperelliptic mapping class groups: we extend the congruence subgroup property to the general class of hyperelliptic mapping class groups introduced above and then determine the centralizers of multitwists and of open subgroups in their profinite completions.

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On the procongruence completion of the Teichmüller modular group

Abstract: For 2g − 2 + n > 0, the Teichmüller modular group Γ g

Cited by 9 publications

References 18 publications

Congruence topologies on the mapping class group

Congruence topologies on the mapping class group

Topics Surrounding the Combinatorial Anabelian Geometry of Hyperbolic Curves Iv: Discreteness and Sections

Notes on hyperelliptic mapping class groups

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