On the equivalence of the definitions of volume of representations

Kim, Sungwoon

doi:10.2140/pjm.2016.280.51

Cited by 9 publications

(15 citation statements)

References 20 publications

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“…For a detailed proof, see [15]. Hence the two different definitions of Francaviglia-Klaff [10] and Bucher-Burger-Iozzi [2] for the volume of a representation of a nonuniform lattice give the same value.…”

Section: Reformulation Of the Volume Of Representationsmentioning

confidence: 89%

“…The authors [16] give a definition of the volume of a representation for a nonuniform lattice in a semisimple Lie group through bounded cohomology, 1 -homology and simplicial volume. In fact, in the case of hyperbolic lattices, it turns out that all definitions of the volume of a representation give the same value [15]. For further details, see [16,Section 6] (a similar proof works for any dimension) and Section 3.…”

Section: Introductionmentioning

confidence: 98%

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On deformation spaces of nonuniform hyperbolic lattices

Kim¹,

Ким²

2016

Math. Proc. Camb. Phil. Soc.

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Let $\Gamma$ be a nonuniform lattice acting on real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected component of the representation variety of $\Gamma$ in SO(n,1). Furthermore, in dimensions 2 and 3, there is a semialgebraic subset of the representation variety such that the volume of a representation is constant on connected components of the semialgebraic subset. Our approach gives a new proof of the local rigidity theorem for nonuniform hyperbolic lattices and the analogue of Soma's theorem, which shows that the number of orientable hyperbolic manifolds dominated by a closed, connected, orientable 3-manifold is finite, for noncompact 3-manifolds.Comment: 25 page

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Section: Reformulation Of the Volume Of Representationsmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 98%

On deformation spaces of nonuniform hyperbolic lattices

Kim¹,

Ким²

2016

Math. Proc. Camb. Phil. Soc.

Self Cite

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“…Firstly, we will recall some definitions on the cohomology group of a topological space with a group action, we refer to [34] and the references therein. Let X be a topological space and G be a group acting continuously on X.…”

Section: 23mentioning

confidence: 99%

“…Define F * alt,b (X, R) as the subspace of F * alt (X, R) consisting of bounded alternating functions. The coboundary operator restricts to the complex F * alt,b (X, R) G and so it defines a cohomology, denoted by H * b (X; G, R), see [16] and also [34,Section 3]. In particular, for a manifold X,…”

Section: 23mentioning

confidence: 99%

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Signature, Toledo invariant and surface group representations in the real symplectic group

Ким¹,

Pansu²,

Wan³

2021

Preprint

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In this paper, by using Atiyah-Patodi-Singer index theorem, we obtain a formula for the signature of a flat symplectic vector bundle over a surface with boundary, which is related to the Toledo invariant of a surface group representation in the real symplectic group and the Rho invariant on the boundary. As an application, we obtain a Milnor-Wood type inequality for the signature. In particular, we give a new proof of the Milnor-Wood inequality for the Toledo invariant in the case of closed surfaces and obtain some modified inequalities for the surface with boundary.

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A Matsumoto–mostow Result for Zimmer’s Cocycles of Hyperbolic Lattices

Moraschini

Savini

2020

Transformation Groups

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Following the philosophy behind the theory of maximal representations, we introduce the volume of a Zimmer’s cocycle Γ × X → PO° (n, 1), where Γ is a torsion-free (non-)uniform lattice in PO° (n, 1), with n > 3, and X is a suitable standard Borel probability Γ-space. Our numerical invariant extends the volume of representations for (non-)uniform lattices to measurable cocycles and in the uniform setting it agrees with the generalized version of the Euler number of self-couplings. We prove that our volume of cocycles satisfies a Milnor–Wood type inequality in terms of the volume of the manifold Γ\ℍn. Additionally this invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to define a family of measurable cocycles with vanishing volume. The same interpretation enables us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map X → PO° (n, 1) with essentially constant sign.As a by-product of our rigidity result for the volume of cocycles, we give a different proof of the mapping degree theorem. This allows us to provide a complete characterization of maps homotopic to local isometries between closed hyperbolic manifolds in terms of maximal cocycles.In dimension n = 2, we introduce the notion of Euler number of measurable cocycles associated to a closed surface group and we show that it extends the classic Euler number of representations. Our Euler number agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. Imitating the techniques developed in the case of the volume, we show a Milnor–Wood type inequality whose upper bound is given by the modulus of the Euler characteristic of the associated closed surface. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, using the interpretation of the Euler number as a multiplicative constant between bounded cohomology classes, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.

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On the equivalence of the definitions of volume of representations

Cited by 9 publications

References 20 publications

On deformation spaces of nonuniform hyperbolic lattices

On deformation spaces of nonuniform hyperbolic lattices

Signature, Toledo invariant and surface group representations in the real symplectic group

A Matsumoto–mostow Result for Zimmer’s Cocycles of Hyperbolic Lattices

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