Some finite quotients of the mapping class group of a surface

Sipe, Patricia L.

doi:10.1090/s0002-9939-1986-0840639-5

Cited by 9 publications

(6 citation statements)

References 6 publications

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“…As explained in [12, §13], 9 a result of Sipe [32] implies that when g ≥ 3 the action of the Torelli group T g on the (2g − 2)nd roots of the canonical bundle factors through the Johnson homomorphism T g → Λ 3 H Z /(θ ∧ H Z ). This implies that K g , and hence all K g,n , act trivially on roots of the canonical bundle when g ≥ 3.…”

Section: In the Category Of Mixed Hodge Structures It Remains Exact A...mentioning

confidence: 99%

Genus 3 mapping class groups are not Kähler

Hain

2014

Journal of Topology

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We prove that finite index subgroups of genus 3 mapping class and Torelli groups that contain the group generated by Dehn twists on bounding simple closed curves are not Kähler. These results are deduced from explicit presentations of the unipotent (aka, Malcev) completion of genus 3 Torelli groups and of the relative completions of genus 3 mapping class groups. The main results follow from the fact that these presentations are not quadratic. To complete the picture, we compute presentations of completed Torelli and mapping class in genera ≥ 4; they are quadratic. We also show that groups commensurable with hyperelliptic mapping class groups and mapping class groups in genera ≤ 2 are not Kähler. ContentsTheorem 4. In genus 3, the Lie algebras t 3 and u 3 are isomorphic to the degree completions of the graded Lie algebras associated to their lower central series:Gr n LCS t 3 and u 3 ∼ = n≥1 Gr n LCS u 3

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Section: In the Category Of Mixed Hodge Structures It Remains Exact A...mentioning

confidence: 99%

Genus 3 mapping class groups are not Kähler

Hain

2014

Journal of Topology

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“…The formulae of Lemma 4 -for the case r = 2, where signs do not matter -where derived earlier by Dabrowski and Percacci [4] by quite involved calculations in local coordinates. Related considerations can also be found in the work of Sipe [18]. She studied r th roots of the unit tangent bundle of hyperbolic surfaces with the aim of describing certain finite quotients of their mapping class group.…”

Section: Claim δ(Wmentioning

confidence: 99%

Generalised spin structures on 2-dimensional orbifolds

Geiges,

Gonzalo

2010

Preprint

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Generalised spin structures, or r-spin structures, on a 2-dimensional orbifold Σ are r-fold fibrewise connected coverings (also called r th roots) of its unit tangent bundle ST Σ. We investigate such structures on hyperbolic orbifolds. The conditions on r for such structures to exist are given. The action of the diffeomorphism group of Σ on the set of r-spin structures is described, and we determine the number of orbits under this action and their size. These results are then applied to describe the moduli space of taut contact circles on left-quotients of the 3-dimensional geometry SL 2 .

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“…In [46] Sipe (and independently Trapp [50]), studied an extension of the symplectic representation. Trapp interpreted the new information explicitly as detecting the action of M g,1,0 on winding numbers of curves on surfaces.…”

Section: ♦24mentioning

confidence: 99%

“…We review what we know about infinite quotients of M. The mapping class group acts naturally on H 1 (S g,b,n ), giving rise to the symplectic representations from M g,1,0 and M g,0,0 to Sp(2g, Z). In [46] Sipe (and independently Trapp [50]), studied an extension of the symplectic representation. Trapp interpreted the new information explicitly as detecting the action of M g,1,0 on winding numbers of curves on surfaces.…”

Section: ♦24mentioning

confidence: 99%

The topology of 3-manifolds, Heegaard distance and the mapping class group of a 2-manifold

Birman¹

2006

Proceedings of Symposia in Pure Mathematics

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We have had a long-standing interest in the way that structure in the mapping class group of a surface reflects corresponding structure in the topology of 3-manifolds, and conversely. We find this area intriguing because the mapping class group (unlike the collection of closed orientable 3-manifolds) is a group which has a rich collection of subgroups and quotients, and they might suggest new ways to approach 3-manifolds. (For example, it is infinite, non-abelian and residually finite [14]). In the other direction, 3-manifolds have deep geometric structure, for example the structure that is associated to intersections between 2-dimensional submanifolds, and that sort of inherently geometric structure might bring new tools to bear on open questions regarding the mapping class group. That dual theme is the focus of this article.In §1 we set up notation and review the background, recalling some of the things that have already been done relating to the correspondence, both ways. We will also describe some important open problems, as we encounter them. We single out for further investigation a new tool which was introduced in [17] by Hempel as a measure of the complexity of a Heegaard splitting of a 3-manifold. His measure of complexity has its origins in the geometry of 3-manifolds. He defined it as the length of the shortest path between certain vertices in the curve complex of a Heegaard surface in the 3-manifold. In fact, the mapping class group acts on the curve complex and the action is faithful. That is, the (extended) mapping class group is isomorphic to the automorphism group of the curve complex. In §2 we will propose some number of open questions which relate to the distance, for study. In §3 we suggest approaches which could be useful in obtaining additional tools to investigate some of the problems posed in §2. Some of the 'additional tools' are in the form of additional open problems.The symbols ♦1, ♦2, ♦3, . . . will be used to highlight known results from the literature.Acknowledgements We thank

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Some finite quotients of the mapping class group of a surface

Cited by 9 publications

References 6 publications

Genus 3 mapping class groups are not Kähler

Genus 3 mapping class groups are not Kähler

Generalised spin structures on 2-dimensional orbifolds

The topology of 3-manifolds, Heegaard distance and the mapping class group of a 2-manifold

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