Volume of Representations and Birationality of Peripheral Holonomy

Guilloux, Antonin

doi:10.1080/10586458.2017.1320240

Cited by 5 publications

(4 citation statements)

References 17 publications

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“…For example both [Dun99] and [KT16] used the properties of the Borel function to prove that the component of the variety X(Γ, P SL(2, C)) containing the holonomy representation of the complete structure on M = Γ\H 3 is birational to its image through the peripheral holonomy map, which is obtained by restricting any representation to the abelian parabolic subgroups determined by the cusps. A similar result has been obtained by [Gui17] for the geometric component of the P SL(n, C)-character variety, but the author needed to conjecture that outside of an analytic neighborhood of the class of the representation π n • i the Borel function is bounded away from its maximum value n+1 3 Vol(Γ\H 3 ). This conjecture could be equivalently stated by saying that any continuous extension of the Borel function to the Parreau-Thurston compactification of X(Γ, P SL(n, C)) has a unique maximum which is not an ideal point (see [Par12] for a definition of the Thurston-Parreau compactification).…”

Section: Introductionsupporting

confidence: 53%

Rigidity at infinity for the Borel function of the tetrahedral reflection lattice

Savini¹

2019

Preprint

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Let Γ be a non-uniform lattice of P SL(2, C). To every representation ρ : Γ → P SL(n, C) it is possible to associate a numerical invariant β n (ρ), called Borel invariant, which is constant on the P SL(n, C)-conjugancy class of the representation ρ and hence defines a function on the character variety X(Γ, P SL(n, C)). This function is continuous with respect to the topology of the pointwise convergence and it satisfies a strong rigidity property: it holds |β n (ρ)| ≤ n+13 Vol(Γ\H 3 ) for every representation ρ : Γ → P SL(n, C) and we have the equality if and only if the representation ρ is conjugated either to π n • i or to π n • i, where i : Γ → P SL(2, C) is the standard lattice embedding and π n : P SL(2, C) → P SL(n, C) is the irreducible representation.We partially answer to a conjecture proposed in [Gui17] when Γ the reflection group associated to a regular ideal tetrahedron. More precisely let Γ 0 < P SL(2, C) be a torsion-free subgroup of Γ and let ρ k : Γ 0 → P SL(n, C) be a sequence of asymptotically maximal representations, that is lim k→∞ β n (ρ k ) = n+13 Vol(Γ 0 \H 3 ). Assume there exists a measurable map ϕ k : P 1 (C) → F (n, C) which is ρ kequivariant. We prove that there must exists a sequence (g k ) k∈N where g k ∈ P SL(n, C) such that lim k→∞ g k ρ k (γ)g −1 k = (π n • i)(γ), for every γ ∈ Γ 0 .ϑ * (B n )(F, G) = B n ((H 0 , w 0 ), . . . , (H 3 , w 3

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Section: Introductionsupporting

confidence: 53%

Rigidity at infinity for the Borel function of the tetrahedral reflection lattice

Savini¹

2019

Preprint

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“…For instance, one could ask if it is possible to extend the ridigity of volume also at ideal points. A similar problem has already been conjectured in [Gui16] relatively to the rigidity of the Borel function with respect to the ideal points of the Morgan-Shalen compactification of the character variety X(Γ, P SL(n, C)). More precisely, the statement is Conjecture 1.1 ( [Gui16]).…”

Section: Introductionmentioning

confidence: 56%

“…A similar problem has already been conjectured in [Gui16] relatively to the rigidity of the Borel function with respect to the ideal points of the Morgan-Shalen compactification of the character variety X(Γ, P SL(n, C)). More precisely, the statement is Conjecture 1.1 ( [Gui16]). Let M be an orientable cusped hyperbolic 3-manifold.…”

Section: Introductionmentioning

confidence: 56%

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

Francaviglia¹,

Savini²

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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Given the fundamental group Γ of a finite-volume complete hyperbolic 3-manifold M , it is possible to associate to any representation ρ : Γ → Isom(H 3 ) a numerical invariant called volume. This invariant is bounded by the hyperbolic volume of M and satisfies a rigidity condition: if the volume of ρ is maximal, then ρ must be conjugated to the holonomy of the hyperbolic structure of M . This paper generalizes this rigidity result by showing that if a sequence of representations of Γ into Isom(H 3 ) satisfies lim n→∞ Vol(ρ n ) = Vol(M ), then there must exist a sequence of elements g n ∈ Isom(H 3 ) such that the representations g n • ρ n • g −1 n converge to the holonomy of M . In particular if the sequence (ρ n ) n∈N converges to an ideal point of the character variety, then the sequence of volumes must stay away from the maximum. In this way we give an answer to [Gui16, Conjecture 1]. We conclude by generalizing the result to the case of k-manifolds and representations in Isom(H m ), where m ≥ k ≥ 3.

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“…Successively, the author extended the same notion to the context of complex and quaternionic lattices getting a stronger rigidity phenomenon. Indeed, as shown by the author and Francaviglia, the volume function is actually rigid also at the ideal points of the character variety [FS18,Sav18], leading to a proof of Guilloux's conjecture [Gui17,Conjecture1] for n = 2.…”

Section: Introductionmentioning

confidence: 91%

Equivariant maps for measurable cocycles with values into higher rank Lie groups

Savini

2019

Preprint

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Let G a semisimple Lie group of non-compact type and let X G be the Riemannian symmetric space associated to it. Suppose X G has dimension n and it does not contain any factor isometric either to R, H 2 or SL(3, R)/SO(3). Given a closed n-dimensional Riemannian manifold N , let Γ = π1(N ) be its fundamental group and Y its universal cover. Consider a representation ρ : Γ → G with a measurable ρ-equivariant map ψ : Y → X G. Connell-Farb described a way to construct a map F : Y → X G which is smooth, ρ-equivariant and with uniformly bounded Jacobian.In this paper we extend the construction of Connell-Farb to the context of measurable cocycles. More precisely, if (Ω, µΩ) is a standard Borel probability Γ-space, let σ : Γ × Ω → G be measurable cocycle which admits a measurable σequivariant map ψ : Y × Ω → X G. We construct a measurable map F : Y × Ω → X G which is σ-equivariant, whose slices are smooth and they have uniformly bounded Jacobian. For such equivariant maps we define also the notion of volume and we prove a sort of mapping degree theorem in this particular context.

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Volume of Representations and Birationality of Peripheral Holonomy

Cited by 5 publications

References 17 publications

Rigidity at infinity for the Borel function of the tetrahedral reflection lattice

Rigidity at infinity for the Borel function of the tetrahedral reflection lattice

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

Equivariant maps for measurable cocycles with values into higher rank Lie groups

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