Galois coverings of moduli spaces of curves and loci of curves with symmetry

Boggi, Marco

doi:10.1007/s10711-012-9821-2

Cited by 12 publications

(49 citation statements)

References 21 publications

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“…This immediately implies the claim that Π(n) = Π(n) C . The statement of the proposition then follows from the short exact sequence (5).…”

Section: Comparison Of C -Congruence Topologiesmentioning

confidence: 93%

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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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“…This immediately implies the claim that Π(n) = Π(n) C . The statement of the proposition then follows from the short exact sequence (5).…”

Section: Comparison Of C -Congruence Topologiesmentioning

confidence: 93%

“…There is an exact sequence (cf. Section 2 in [5]. Note that here we are considering pure mapping class groups):…”

Section: Centralizers Of Pro-c Multitwistsmentioning

confidence: 99%

Congruence topologies on the mapping class group

Boggi

2020

Journal of Algebra

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“…For suitable choices of the finite group

G

, the analytic space

_{G} {\bar{M}}_{g}^{a n}

is a compact smooth complex manifold (see ). Moreover, the map forgetting the level structure

_{G} {\bar{M}}_{g}^{a n} \to {\bar{M}}_{g}^{a n}

is a Galois covering ramified along the divisor at infinity.…”

Section: Virtually Kähler Groupsmentioning

confidence: 99%

“…In order to prove Theorem we will compute

π_{1} (_{G} {\bar{scriptM}}_{g}^{a n} false)

in such cases. For the sake of conciseness, most arguments borrowed from are only sketched below.…”

Section: Virtually Kähler Groupsmentioning

confidence: 99%

“…Remark The same method yields (many) other virtually Kähler quotients of

{Mod}_{g}

. For instance we could take for odd

p

the normal subgroup generated by

p

th powers of separating bounded simple closed curves of genus 1 and the

3 p

th powers of other Dehn twists, see ([, Proposition 2.8],

k = 7)

and also [, Proposition 3.12] for other possibilities.…”

Section: Virtually Kähler Groupsmentioning

confidence: 99%

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Quotients of the mapping class group by power subgroups

Aramayona

Funar

2019

Bull. London Math. Soc.

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We study the quotient of the mapping class group Modgn of a surface of genus g with n punctures, by the subgroup prefixModgnfalse[pfalse] generated by the pth powers of Dehn twists. Our first main result is that prefixModg1/prefixModg1false[pfalse] contains an infinite normal subgroup of infinite index, and in particular is not commensurable to a higher rank lattice, for all but finitely many explicit values of p. Next, we prove that prefixModg0/prefixModg0false[pfalse] contains a Kähler subgroup of finite index, for every p⩾2 coprime with six. Finally, we observe that the existence of finite‐index subgroups of Modg0 with infinite abelianization is equivalent to the analogous problem for prefixModg0/prefixModg0false[pfalse].

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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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Galois coverings of moduli spaces of curves and loci of curves with symmetry

Cited by 12 publications

References 21 publications

Congruence topologies on the mapping class group

Congruence topologies on the mapping class group

Quotients of the mapping class group by power subgroups

Notes on hyperelliptic mapping class groups

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