Constructing equivariant maps for representations

Francaviglia, Stefano

doi:10.5802/aif.2434

Cited by 8 publications

(17 citation statements)

References 32 publications

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“…Given a continuous map f : M 1 → M 2 of non-vanishing degree, thanks to the work of [BM96,Fra09] there exists an essentially unique measurable map f : ∂H n → ∂H n which is π 1 (f )-equivariant. Now if we assume that σ admits a boundary map φ : S n−1 × X → S n−1 , the pullback cocycle along f admits the following boundary map:…”

Section: Maximal Cocycles and Mapping Degreementioning

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: Maximal Cocycles and Mapping Degreementioning

confidence: 99%

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

Moraschini

Savini

2020

Transformation Groups

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“…Given a lattice Γ ≤ G(n), there always exists a Patterson-Sullivan density associated to it. Moreover, it is essentially unique by the doubly ergodic action of Γ on the boundary at infinity ∂ ∞ H n K (see, for instance, [49,41,14,43,24]). It is worth mentioning that the construction of the Patterson-Sullivan density has been extended by Albuquerque [2,3] to higher rank lattices in a semisimple Lie group G of noncompact type.…”

Section: A Savinimentioning

confidence: 99%

“…Proof. Since ρ is non-elementary, by [14,24] there exists a measurable boundary map ϕ : The previous map allows one to define an essentially unique boundary map associated to σ ρ as follows:…”

Section: A Savinimentioning

confidence: 99%

Natural Maps for Measurable Cocycles of Compact Hyperbolic Manifolds

Savini

2021

J. Inst. Math. Jussieu

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Let $\operatorname {\mathrm {{\rm G}}}(n)$ be equal to either $\operatorname {\mathrm {{\rm PO}}}(n,1),\operatorname {\mathrm {{\rm PU}}}(n,1)$ or $\operatorname {\mathrm {\textrm {PSp}}}(n,1)$ and let $\Gamma \leq \operatorname {\mathrm {{\rm G}}}(n)$ be a uniform lattice. Denote by $\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}}$ the hyperbolic space associated to $\operatorname {\mathrm {{\rm G}}}(n)$ , where $\operatorname {\mathrm {{\rm K}}}$ is a division algebra over the reals of dimension d. Assume $d(n-1) \geq 2$ . In this article we generalise natural maps to measurable cocycles. Given a standard Borel probability $\Gamma $ -space $(X,\mu _X)$ , we assume that a measurable cocycle $\sigma :\Gamma \times X \rightarrow \operatorname {\mathrm {{\rm G}}}(m)$ admits an essentially unique boundary map $\phi :\partial _\infty \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \times X \rightarrow \partial _\infty \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ whose slices $\phi _x:\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ are atomless for almost every $x \in X$ . Then there exists a $\sigma $ -equivariant measurable map $F: \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \times X \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ whose slices $F_x:\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ are differentiable for almost every $x \in X$ and such that $\operatorname {\mathrm {\textrm {Jac}}}_a F_x \leq 1$ for every $a \in \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}}$ and almost every $x \in X$ . This allows us to define the natural volume $\operatorname {\mathrm {\textrm {NV}}}(\sigma )$ of the cocycle $\sigma $ . This number satisfies the inequality $\operatorname {\mathrm {\textrm {NV}}}(\sigma ) \leq \operatorname {\mathrm {\textrm {Vol}}}(\Gamma \backslash \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}})$ . Additionally, the equality holds if and only if $\sigma $ is cohomologous to the cocycle induced by the standard lattice embedding $i:\Gamma \rightarrow \operatorname {\mathrm {{\rm G}}}(n) \leq \operatorname {\mathrm {{\rm G}}}(m)$ , modulo possibly a compact subgroup of $\operatorname {\mathrm {{\rm G}}}(m)$ when $m>n$ . Given a continuous map $f:M \rightarrow N$ between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context.

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“…The Patterson-Sullivan family of measures and the BCGnatural map. For more details about the following definitions and constructions we recomend the reader to see the first sections of [Fra09]. Let Γ < Isom(H k ) be a discrete group of divergence type, that is a subgroup for which the Poincaré series diverges at the critical exponent δ(Γ).…”

Section: Preliminary Definitionsmentioning

confidence: 99%

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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Constructing equivariant maps for representations

Cited by 8 publications

References 32 publications

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

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

Natural Maps for Measurable Cocycles of Compact Hyperbolic Manifolds

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

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