On deformation spaces of nonuniform hyperbolic lattices

Kim, Sungwoon; Ким, Инканг

doi:10.1017/s0305004116000293

Cited by 7 publications

(4 citation statements)

References 24 publications

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“…Moreover, the second author also showed that the rigidity holds for complex and quaternionic lattices [Sav20]). The interest in the study of volume of representations has recently grown, leading to the development of a rich literature [Poz15], [KK16], [Tho18], [Far], [Far20].…”

Section: Historical Backgroundmentioning

confidence: 99%

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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Section: Historical Backgroundmentioning

confidence: 99%

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

Moraschini

Savini

2020

Transformation Groups

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“…where [M] is the generator of H n (M, R) ∼ = H n (Γ, R) given by the fundamental class and ρ * (ω n ) ∈ H n (Γ, R) is the pullback of ω n . This definition has been extended to the non-compact case by various authors [Dun99, Fra04, BIW10, KK12a, BBI13], and the equivalence of these definitions has been recently established in [KK13]. We will use the cohomological definition introduced in [BBI13], which parallels the definition for surface groups given in [BIW10], and we briefly recall it here (for more details see § 4).…”

Section: Introductionmentioning

confidence: 99%

“…If M is non-compact, the situation parallels the one above, at least in high dimension. In fact, using an approach via Schläfli's formula as in [BCG07], Kim and Kim proved that if M is a finite volume hyperbolic manifold of dimension ≥ 4, the volume is constant on the connected components of Hom(π 1 (M), Isom + (H n )) [KK13]. Like in the compact case, in odd dimension the nature of these values is mysterious.…”

Section: Introductionmentioning

confidence: 99%

Integrality of Volumes of Representations

Bucher¹,

Burger²,

Iozzi³

2014

Preprint

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Let M be an oriented complete hyperbolic n-manifold of finite volume. Using the definition of volume of a representation previously given by the authors in [BBI13] we show that the volume of a representation ρ :Moreover we prove that the volume is continuous in all dimension and hence, if dim M = 2m ≥ 4, it is constant on connected components of the representation variety.If M is not compact and dim M = 3, the volume is not locally constant and we give explicit examples of representations with volume as arbitrary as the volume of hyperbolic manifolds obtained from M via Dehn fillings.

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“…In a more general setting, Francaviglia and Klaff [FK06] proved some similar rigidity results for their definition of volume of a representation Γ → PO(m, 1), this time assuming m ≥ n ≥ 3 (the rigidity of volume actually holds also at infinity, as proved by Francaviglia and the second author [FS18] for the real hyperbolic lattices: Moreover, the second author also showed that the rigidity holds for complex and quaternionic lattices [Sav18]). The interest in the study of volume of representations has recently grown, leading to the development of a rich literature [Poz15,KK16,Tho18,Fara,Farb].…”

mentioning

confidence: 99%

A Matsumoto-Mostow result for Zimmer's cocycles of hyperbolic lattices

Moraschini,

Savini

2019

Preprint

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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\H n . 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 deformation spaces of nonuniform hyperbolic lattices

Cited by 7 publications

References 24 publications

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

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

Integrality of Volumes of Representations

A Matsumoto-Mostow result for Zimmer's cocycles of hyperbolic lattices

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