Volume rigidity ad ideal points of the character variety of hyperbolic 3-manifolds

Francaviglia, Stefano; Savini, Alessio

doi:10.2422/2036-2145.201709_010

Cited by 9 publications

(9 citation statements)

References 24 publications

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“…This is a suitable adaptation of Mostow-Prasad rigidity to the context of representations. 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 [Sav20]).…”

Section: Historical Backgroundmentioning

confidence: 66%

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: 66%

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

Moraschini

Savini

2020

Transformation Groups

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“…They also show that the volume is rigid, that means |Vol(ρ)| ≤ Vol(Γ\H n ) and the equality holds if and only if ρ is conjugated to the standard lattice embedding, which is an equivalent formulation of Mostow rigidity. For sake of completeness we have to say that actually a similar result was proved for representations into PO(m, 1) with m ≥ n ≥ 3 by [FK06] and it remains valid even at ideals points of the PO(m, 1)character variety (see [FS18] for the case of real lattices and [Sav18] for the cases of complex and quaternionic lattices).…”

Section: Introductionmentioning

confidence: 59%

Borel invariant for measurable cocycles of 3-manifold groups

Savini¹

2019

Preprint

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Let Γ be a non-uniform lattice of PSL(2, C). Given any representation ρ : Γ → PSL(n, C) we can define numerical invariant β n (ρ), called Borel invariant, which remains constant along the PSL(n, C)-conjugancy class of ρ. Additionally this invariant is rigid in the following sense: it holds |β n (ρ)| ≤ n+13 Vol(Γ\H 3 ) for every representation ρ : Γ → PSL(n, C) and the equality is attained if and only if the representation ρ is conjugated either toWe extend the notion of Borel invariant to the more general setting of Zimmer's theory of cocycles. More precisely, let (X, µ X ) be a probability space on which Γ acts in a measure-preserving way. Let σ : Γ × X → PSL(n, C) be a measurable cocycle and assume that φ : P 1 (C) × X → F (n, C) is a measurable σ-equivariant map, that means φ(γξ, γx) = σ(γ, x)φ(ξ, x) for every γ ∈ Γ and almost every (ξ, x) ∈ P 1 (C) × X. We define the notion of Borel invariant β n (σ) associated to the cocycle σ and we prove that it holds the same bound which is valid in the classic case. Additionally we show that if the invariant is maximal then the cocycle must be cohomologous either to the one associated to the irreducible representation π n or to its complex conjugated.

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“…Some of these results rely on the notion of natural maps introduced by Besson, Courtois and Gallot [BCG95, BCG96, BCG98] and more precisely on the sharp estimate on the Jacobian of such maps. Actually the volume rigidity can be extended even at ideal points of the representations space, as proved by both Francaviglia and one of the authors [FS18,Sav18].…”

Section: Introductionmentioning

confidence: 84%

Multiplicative constants and maximal measurable cocycles in bounded cohomology

Moraschini,

Savini

2019

Preprint

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Inspired by the theory of maximal representations via bounded cohomology, we introduce the notion of both multiplicative constant and maximal measurable cocycle. Maximal cocycles satisfy usually a trivialization property, since under suitable hypothesis they are cohomologous to a preferred representation.The main application of this paper is the definition and the study of the Cartan invariant of a measurable cocycle associated to a complex hyperbolic lattice. We first extend the classic Cartan invariant of representations to measurable cocycles. Then, built on our fibered multiplicative formula, we completely describe the relation between totally real cocycles and the vanishing of the Cartan invariant. In this way we get an extension of a result of Burger and Iozzi [BI12, Theorem 1.1] to the case of measurable cocycles. Finally, applying our theory of multiplicative constants, we completely characterize measurable cocycles with maximal Cartan invariant as the ones which can be trivialized. Contents 3.3. Easy multiplicative formula 25 3.4. Multiplicative constants and maximal measurable cocycles 26 3.5. Applications of the easy multiplicative formula 29 4. Multiplicative constants and change of coefficients 33 4.1. Fibered pullback maps 34 4.2. Factorization of fibered pullback maps and multiplicative constants 36 5. Cartan invariant of measurable cocycles of complex hyperbolic lattices 40 5.1. Cartan invariant of measurable cocycles 40 5.2. The Cartan invariant as a multiplicative constant 44 References 52

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Volume rigidity ad ideal points of the character variety of hyperbolic 3-manifolds

Cited by 9 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

Borel invariant for measurable cocycles of 3-manifold groups

Multiplicative constants and maximal measurable cocycles in bounded cohomology

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