Simplicial volume of locally symmetric spaces covered by $${\rm SL}_{3}{\mathbb{R}}/{\rm SO}(3)$$

Bucher-Karlsson, Michelle

doi:10.1007/s10711-007-9158-4

Cited by 32 publications

(26 citation statements)

References 7 publications

(34 reference statements)

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“…in general is quite mysterious, but in the cases where it is known, their properties are complementary to each other: -In degree 1 the map h (1) Γ is injective, whereas the image of c …”

Section: It Is a Very Interesting Open Question Whether The Comparisomentioning

confidence: 99%

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Remarks On and Around Bounded Differential Forms

Wienhard¹

2012

Pure and Applied Mathematics Quarterly

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“…in general is quite mysterious, but in the cases where it is known, their properties are complementary to each other: -In degree 1 the map h (1) Γ is injective, whereas the image of c …”

Section: It Is a Very Interesting Open Question Whether The Comparisomentioning

confidence: 99%

“…It is tempting to ask the following [16], and by Bucher [1] in the case when G is locally isomorphic to SL(3, R), which is excluded in [16].…”

Section: It Is a Very Interesting Open Question Whether The Comparisomentioning

confidence: 99%

Remarks On and Around Bounded Differential Forms

Wienhard¹

2012

Pure and Applied Mathematics Quarterly

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“…The Γ-action on X extends continuously to X ∪ X(∞), and the stabilizers of the boundary points in G are exactly the proper parabolic subgroups of G. X also admits several other compactifications such as the Satake compactifications on which the Γ-action extends continuously, and the stabilizers of the boundary points in G are smaller than the parabolic subgroups of G in general. [251] and [83] for the definition and the precise statement of the result). On the other hand, if the Q-rank of Γ\X is at least 3, then the simplicial volume of Γ\X is zero.…”

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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“…Actually, the case of the symmetric space of SL 3 (R) is not covered by [100]; it was claimed in [134], but the proof therein is incomplete. However a proof is provided by M. Bucher-Karlsson in [13]. M. Gromov has conjectured more generally that closed manifolds with non-positive curvature and negative Ricci curvature have positive simplicial volume.…”

Section: Refinements Of Ordinary Cohomology and Numerical Boundsmentioning

confidence: 99%