The simplicial volume of closed manifolds covered by ℍ<sup>2</sup>× ℍ<sup>2</sup>

Bucher-Karlsson, Michelle

doi:10.1112/jtopol/jtn012

Cited by 43 publications

(53 citation statements)

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“…When the Q-rank of G is equal to 1 or 2, it is not known whether the simplicial volume of Γ\X is positive or not. For some Q-rank 1 locally symmetric spaces including the Hilbert modular varieties, it is known that the simplicial volume is positive [262], and for the Hilbert modular surfaces, the simplicial volume can be computed explicitly using the result in [84].…”

Section: Properties Of Actions Of Arithmetic Groups γ Onmentioning

confidence: 99%

A tale of two groups: arithmetic groups and mapping class groups

2012

Handbook of Teichmüller Theory, Volume III

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SummaryIn this chapter, we discuss similarities, differences and interaction between two natural and important classes of groups: arithmetic subgroups Γ of Lie groups G and mapping class groups Mod g,n of surfaces of genus g with n punctures. We also mention similar properties and problems for related groups such as outer automorphism groups Out(F n ), Coxeter groups and hyperbolic groups. Since groups are often effectively studied by suitable spaces on which they act, we also discuss related properties of actions of arithmetic groups on symmetric spaces and actions of mapping class groups on Teichmüller spaces, hoping to get across the point that it is the existence of actions on good spaces that makes the groups interesting and special, and it is also the presence of large group actions that also makes the spaces interesting. Interaction between locally symmetric spaces and moduli spaces of Riemann surfaces through the example of the Jacobian map will also be discussed in the last part of this chapter. Since reduction theory, i.e., finding good fundamental domains for proper actions of discrete groups, is crucial to transformation group theory, i.e., to understand the algebraic structures of groups, properties of group actions and geometry, topology and compactifications of the quotient spaces, we discuss many different approaches to reduction theory of arithmetic groups acting on symmetric spaces. These results for arithmetic groups motivate some results on fundamental domains for the action of mapping class groups on Teichmüller spaces. For example, the Minkowski reduction theory of quadratic forms is generalized to the action of Mod g = Mod g,0 on the Teichmüller space T g to construct an intrinsic fundamental domain consisting of finitely many cells, * Partially Supported by NSF grant DMS 0905283 2 solving a weaker version of a folklore conjecture in the theory of Teichmüller spaces.On several aspects, more results are known for arithmetic groups, and we hope that discussion of results for arithmetic groups will suggest corresponding results for mapping class groups and hence increase interactions between the two classes of groups. In fact, in writing this survey and following the philosophy of this chapter, we noticed the natural procedure in §5.12 of constructing the Deligne-Mumford compactification of the moduli space of Riemann surfaces from the Bers compactification of the Teichmüller space by applying the general procedure of Satake compactifications of locally symmetric spaces, i.e., how to pass from a compactification of a symmetric space to a compactification of a locally symmetric space by making use of the reduction theory for arithmetic groups.The layout of this chapter is as follows. In §1, the introduction, we discuss some general questions about discrete groups, group actions and transformation group theories. In §2, we summarize results on arithmetic subgroups Γ of semisimple Lie groups G and mapping class groups Mod g,n of surfaces of genus g with n punctures, and their actions on symmetri...

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Section: Properties Of Actions Of Arithmetic Groups γ Onmentioning

confidence: 99%

A tale of two groups: arithmetic groups and mapping class groups

2012

Handbook of Teichmüller Theory, Volume III

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“…The combinatorial volume cocycle Vol 4 used for the proofs of Theorems 1 and 2 already appears in [2]: Its sup norm Vol 4 ∞ = 2/3 as a cohomology class in H 4 c (P SL(2, R)×P SL(2, R)) relates the simplicial volume of products of surfaces to the simplicial volume of their factors:…”

Section: Introductionmentioning

confidence: 99%

On Minimal Triangulations of Products of Convex Polygons

Bucher-Karlsson

2008

Discrete Comput Geom

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“…The non-vanishing part of the above corollary is due to [25,5,20]. After posting this paper, the author was kindly informed by Pablo Suárez-Serrato that the Corollary 1.4 has been obtained by him in [52].…”

Section: Introductionmentioning

confidence: 99%

Geometric structures, Gromov norm and Kodaira dimensions

Zhang

2017

Advances in Mathematics

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A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP URL' above for details on accessing the published version and note that access may require a subscription.For more information, please contact the WRAP Team at: wrap@warwick.ac.uk GEOMETRIC STRUCTURES, GROMOV NORM AND KODAIRA DIMENSIONS WEIYI ZHANGAbstract. We define the Kodaira dimension for 3-dimensional manifolds through Thurston's eight geometries, along with a classification in terms of this Kodaira dimension. We show this is compatible with other existing Kodaira dimensions and the partial order defined by non-zero degree maps. IntroductionComplex Kodaira dimension κ h (M, J) provides a very successful classification scheme for complex manifolds. This notion is generalized by several authors (c.f. [31,39,40,26,27]) to symplectic manifolds, especially of dimension two and four. In these two dimensions, this symplectic Kodaira dimension is independent of the choice of symplectic structures [31]. In other words, it is a smooth invariant of the manifold which is thus denoted by κ s (M ). In dimension four, the smaller the symplectic Kodaira dimension, the more we know. Symplectic 4-manifolds with κ s = −∞ are diffeomorphic to rational or ruled surfaces [36]. When κ s = 0, all known examples are K3 surface, Enrique surface and T 2 bundles over T 2 . Moreover, it is shown in [31] that a symplectic manifold with κ s = 0 has the same homological invariants as one of the manifolds listed above. When κ s = 1 or 2, no classification is possible since symplectic manifolds in both categories could admit arbitrary finitely presented group as their fundamental group [19].In [9], the authors prove that complex and symplectic Kodaira dimensions are compatible with each other. More precisely, when a 4-manifold M admits at the same time both complex and symplectic structures (but the structures are not necessarily compatible with each other), then κ s (M ) = κ h (M, J). In [35], a general framework of "additivity of Kodaira dimension" is provided to further understand the compatibility of various Kodaira dimensions in possibly different dimensions. In particular, it is shown that the Kodaira dimensions are additive for fiber bundles, Lefschetz fibrations and coverings.Higher dimensional generalizations of Kodaira dimension, e.g. symplectic Kodaira dimension in dimensions six or higher, are less understood except for a proposed definition in [33]. Like complex Kodaira dimension, it will no longer be a smooth invariant. Hence, the study of this notion in higher dimensions will be associated to the study of deformation classes of symplectic structures and symplectic birational geometry.As suggested by the additivity framework, dimension three should also attach certain counterpart of Kodaira dimension. In this paper, we give a definition of Kodaira dimension κ t (M ) in dimension three through Thurston's eight ...

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The simplicial volume of closed manifolds covered by ℍ²× ℍ²

Cited by 43 publications

References 9 publications

A tale of two groups: arithmetic groups and mapping class groups

A tale of two groups: arithmetic groups and mapping class groups

On Minimal Triangulations of Products of Convex Polygons

Geometric structures, Gromov norm and Kodaira dimensions

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The simplicial volume of closed manifolds covered by ℍ2× ℍ2

Cited by 43 publications

References 9 publications

A tale of two groups: arithmetic groups and mapping class groups

A tale of two groups: arithmetic groups and mapping class groups

On Minimal Triangulations of Products of Convex Polygons

Geometric structures, Gromov norm and Kodaira dimensions

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Product

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The simplicial volume of closed manifolds covered by ℍ²× ℍ²