The fine structure of the moduli space of abelian differentials in genus 3

Looijenga, Eduard; Mondello, Gabriele

doi:10.1007/s10711-013-9845-2

Cited by 20 publications

(20 citation statements)

References 8 publications

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“…Remark 4.7. The symmetric mapping class group SMod(π * ) is an infinite-index subgroup of Mod(π * ) corresponding to the orbifold fundamental group 8 of projectivized hyperelliptic connected components of the moduli spaces of translation surfaces: see, e.g., Looijenga-Mondello [10]. Thus, we have just shown that, in a certain sense, the hyperelliptic Rauzy diagrams "see" the topology of the projectivized hyperelliptic connected components of the moduli spaces of translation surfaces.…”

Section: 22mentioning

confidence: 99%

Zorich conjecture for hyperelliptic Rauzy–Veech groups

2017

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Section: 22mentioning

confidence: 99%

Zorich conjecture for hyperelliptic Rauzy–Veech groups

2017

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“…We stress that Theorem C is already non-optimal in genus 2, since PΩM ′ 2 (2) and PΩM ′ 2 (1, 1) are affine, and in genus 3, since PΩM ′ 3 (4) and PΩM ′ 3 (3, 1) are affine (see [25]). As an extension of Looijenga's question, it seems natural to wonder the following.…”

Section: 42mentioning

confidence: 99%

On the cohomological dimension of the moduli space of Riemann surfaces

Mondello

2017

Duke Math. J.

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The moduli space of Riemann surfaces of genus g >= 2 is (up to a finite etale cover) a complex manifold, so it makes sense to speak of its Dolbeault cohomological dimension. The conjecturally optimal bound is g - 2. This expectation is verified in low genus and is supported by Harer's computation of its de Rham cohomological dimension and by vanishing results in the tautological intersection ring. In this article, we prove that such a dimension is at most 2g - 2. We also prove an analogous bound for the moduli space of Riemann surfaces with marked points. The key step is to show that the Dolbeault cohomological dimension of each stratum of translation surfaces is at most g. In order to do that, we produce an exhaustion function whose complex Hessian has controlled index: the construction of such a function relies on some basic geometric properties of translation surfaces

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“…The conjecture says that each connected component of M d has homotopy type K(G, 1), where G is a group commensurable to some mapping class group. Hamenstädt uses the results in (Looijenga & Mondello, 2014)…”

Section: Introductionmentioning

confidence: 99%

On parabolic subgroups of Artin–Tits groups of spherical type

Cumplido¹,

Gebhardt²,

González-Meneses³

et al. 2019

Advances in Mathematics

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Let A and A be two Artin groups of spherical type, and let A 1 , . . . , A p (resp. A 1 , . . . , A q ) be the irreducible components of A (resp. A ). We show that A and A are commensurable if and only if p = q and, up to permutation of the indices, A i and A i are commensurable for every i. We prove that, if two Artin groups of spherical type are commensurable, then they have the same rank. For a fixed n, we give a complete classification of the irreducible Artin groups of rank n that are commensurable with the group of type A n . Note that it will remain 6 pairs of groups to compare to get the complete classification of Artin groups of spherical type up to commensurability. Proposition 2. Let Γ and Ω be two Coxeter graphs of spherical type. If A[Γ] and A[Ω] are commensurable, then Γ and Ω have the same number of vertices. Proof. Suppose that A[Γ] and A[Ω] are commensurable. Let n be the number of vertices of Γ and let m be the number of vertices of Ω. We know that the cohomological dimension of A[Γ] is n and the cohomological dimension of A[Ω] is m (Paris, 2004, Proposition 3.1). As every finite index subgroup of A[Γ] has the same cohomological dimension as A[Γ] and every finite index subgroup of A[Ω] has the same cohomological dimension as A[Ω], we have n = m.

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The fine structure of the moduli space of abelian differentials in genus 3

Cited by 20 publications

References 8 publications

Zorich conjecture for hyperelliptic Rauzy–Veech groups

Zorich conjecture for hyperelliptic Rauzy–Veech groups

On the cohomological dimension of the moduli space of Riemann surfaces

On parabolic subgroups of Artin–Tits groups of spherical type

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