Simplical volume of closed locally symmetric spaces of non-compact type

Lafont, Jean-François; Schmidt, Benjamin

doi:10.1007/s11511-006-0009-1

Cited by 80 publications

(79 citation statements)

References 11 publications

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“…It follows from [LaSch06] that ω ∞ < ∞. Indeed, Lafont and Schmidt answer affirmatively the conjecture of Gromov that the simplicial volume of a closed locally symmetric space of noncompact type is strictly positive (relying on [Gr82] and [Bu05] in the cases where the universal cover of M has an SL(2, R)/SO(2) or SL(3, R)/SO(3) factor respectively).…”

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confidence: 88%

The proportionality constant for the simplicial volume of locally symmetric spaces

Bucher-Karlsson

2008

Colloq. Math.

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Abstract. We follow ideas going back to Gromov's seminal article [Publ. Math. IHES 56 (1982)] to show that the proportionality constant relating the simplicial volume and the volume of a closed, oriented, locally symmetric space M = Γ \G/K of noncompact type is equal to the Gromov norm of the volume form in the continuous cohomology of G. The proportionality constant thus becomes easier to compute. Furthermore, this method also gives a simple proof of the proportionality principle for arbitrary manifolds.

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confidence: 88%

The proportionality constant for the simplicial volume of locally symmetric spaces

Bucher-Karlsson

2008

Colloq. Math.

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“…Take a product of three non-compact, locally symmetric space of finite volume. Its simplicial volume vanishes, but on the other hand there always exists a compact locally symmetric space with isometric universal cover [1] and the simplicial volume of closed locally symmetric spaces of non-compact type is known to be nonzero [7].…”

Section: Theorem 12 ([4] [11]) Let M and N Be Two Compact Riemannimentioning

confidence: 99%

“…Theorem 1.6 (Degree theorem, [3,7,10]). For every n ∈ N there is a constant C n > 0 with the following property: Let M be an n-dimensional locally symmetric space of non-compact type with finite volume.…”

Section: Theorem 14 (Proportionality Principle) Let M and N Be Two mentioning

confidence: 99%

Piecewise straightening and Lipschitz simplicial volume

Strzałkowski

2017

J. Topol. Anal.

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We study the Lipschitz simplicial volume, which is a metric version of the simplicial volume. We introduce the piecewise straightening procedure for singular chains, which allows us to generalize the proportionality principle and the product inequality to the case of complete Riemannian manifolds of finite volume with sectional curvature bounded from above. We obtain also yet another proof of the proportionality principle in the compact case by a direct approximation of the smearing map.

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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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Simplical volume of closed locally symmetric spaces of non-compact type

Cited by 80 publications

References 11 publications

The proportionality constant for the simplicial volume of locally symmetric spaces

The proportionality constant for the simplicial volume of locally symmetric spaces

Piecewise straightening and Lipschitz simplicial volume

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

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