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Savini Alessio - One of the best experts on this subject based on the ideXlab platform.
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A Matsumoto-Mostow result for Zimmer's cocycles of hyperbolic lattices
'Springer Science and Business Media LLC', 2020Co-Authors: Moraschini Marco, Savini AlessioAbstract:As for the theory of maximal representations, we introduce the volume of a Zimmer's cocycle $\Gamma \times X \rightarrow \mbox{PO}^\circ(n, 1)$, where $\Gamma$ is a torsion-free (non-)uniform lattice in $\mbox{PO}^\circ(n, 1)$, with $n \geq 3$, and $X$ is a suitable standard Borel probability $\Gamma$-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 $\Gamma \backslash \mathbb{H}^n$. This invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a Measurable Map $X \rightarrow \mbox{PO}(n, 1)$ with essentially constant sign. As a by-product of our rigidity result for the volume of cocycles, we give a new 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 closed surface groups. It extends the classic Euler number of representations and it agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. We show a Milnor-Wood type inequality whose upper bound is given by the modulus of the Euler characteristic. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.Comment: 54 pages, we fixed some typos following referees' suggestions. Moreover, we rewrote Section 3 and we modified Corollary 5.14. To appear in Transform. Group
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Equivariant Maps for Measurable cocycles with values into higher rank Lie groups
2020Co-Authors: Savini AlessioAbstract:Let $G$ a semisimple Lie group of non-compact type and let $\mathcal{X}_G$ be the Riemannian symmetric space associated to it. Suppose $\mathcal{X}_G$ has dimension $n$ and it has no factor isometric either to $\mathbb{R},\mathbb{H}^2$ or $\text{SL}(3,\mathbb{R})/\text{SO}(3)$. Given a closed $n$-dimensional Riemannian manifold $N$, let $\Gamma=\pi_1(N)$ be its fundamental group and $Y$ its universal cover. Consider a representation $\rho:\Gamma \rightarrow G$ with a Measurable $\rho$-equivariant Map $\psi:Y \rightarrow \mathcal{X}_G$. Connell-Farb described a way to construct a Map $F:Y\rightarrow \mathcal{X}_G$ which is smooth, $\rho$-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 $(\Omega,\mu_\Omega)$ is a standard Borel probability $\Gamma$-space, let $\sigma:\Gamma \times \Omega \rightarrow G$ be Measurable cocycle which admits a Measurable $\sigma$-equivariant Map $\psi:Y \times \Omega \rightarrow \mathcal{X}_G$. We construct a Measurable Map $F: Y \times \Omega \rightarrow \mathcal{X}_G$ which is $\sigma$-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.Comment: 17 pages; New statements obtained substituting a lattice with the fundamental group of a closed Riemannian manifold + References adde
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Natural Maps for Measurable cocycles of compact hyperbolic manifolds
2019Co-Authors: Savini AlessioAbstract:Let $\text{G}(n)$ be equal either to $\text{PO}(n,1),\text{PU}(n,1)$ or $\text{PSp}(n,1)$ and let $\Gamma \leq \text{G}(n)$ be a uniform lattice. Denote by $\mathbb{H}^n_K$ the hyperbolic space associated to $\text{G}(n)$, where $K$ is a division algebra over the reals of dimension $d=\dim_{\mathbb{R}} K$. Assume $d(n-1) \geq 2$. In this paper we define the notion of natural Map in the setting of Zimmer's cocycles theory. More precisely, let $(X,\mu_X)$ be a standard Borel probability $\Gamma$-space without atoms. Assume that a Measurable cocycle $\sigma:\Gamma \times X \rightarrow \text{G}(m)$ admits an essentially unique boundary Map $\phi:\partial_\infty \mathbb{H}^n_K \times X \rightarrow \partial_\infty \mathbb{H}^m_K$ whose slices $\phi_x:\mathbb{H}^n_K \rightarrow \mathbb{H}^m_K$ are essentially injective for almost every $x \in X$. Then there exists a $\sigma$-equivariant Measurable Map $F: \mathbb{H}^n_K \times X \rightarrow \mathbb{H}^m_K$ whose slices $F_x:\mathbb{H}^n_K \rightarrow \mathbb{H}^m_K$ are differentiable for almost every $x \in X$ and such that $\text{Jac}_a F_x \leq 1$ for every $a \in \mathbb{H}^n_K$ and almost every $x \in X$. The previous properties allow us to define the natural volume $\text{NV}(\sigma)$ of the cocycle $\sigma$. This number is constant along the $\text{G}(m)$-cohomology class of $\sigma$ and it satisfies the Milnor-Wood type inequality $\text{NV}(\sigma) \leq \text{Vol}(\Gamma \backslash \mathbb{H}^n_K)$. Additionally the equality holds if and only if $\sigma$ is cohomologous to the cocycle induced by the standard lattice embedding $i:\Gamma \rightarrow \text{G}(n) \leq \text{G}(m)$. 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 and a characterization of Maps homotopic to local isometries in terms of maximal cocycles.Comment: 27 page
Rafael D Sorkin - One of the best experts on this subject based on the ideXlab platform.
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discreteness without symmetry breaking a theorem
Modern Physics Letters A, 2009Co-Authors: Luca Bombelli, Joe Henson, Rafael D SorkinAbstract:This paper concerns random sprinklings of points into Minkowski spacetime (Poisson processes). It proves that there exists no equivariant Measurable Map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to "Lorentz breaking" effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance.
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discreteness without symmetry breaking a theorem
arXiv: General Relativity and Quantum Cosmology, 2006Co-Authors: Luca Bombelli, Joe Henson, Rafael D SorkinAbstract:This paper concerns sprinklings into Minkowski space (Poisson processes). It proves that there exists no equivariant Measurable Map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to ``Lorentz breaking'' effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance.
Joe Henson - One of the best experts on this subject based on the ideXlab platform.
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discreteness without symmetry breaking a theorem
Modern Physics Letters A, 2009Co-Authors: Luca Bombelli, Joe Henson, Rafael D SorkinAbstract:This paper concerns random sprinklings of points into Minkowski spacetime (Poisson processes). It proves that there exists no equivariant Measurable Map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to "Lorentz breaking" effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance.
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discreteness without symmetry breaking a theorem
arXiv: General Relativity and Quantum Cosmology, 2006Co-Authors: Luca Bombelli, Joe Henson, Rafael D SorkinAbstract:This paper concerns sprinklings into Minkowski space (Poisson processes). It proves that there exists no equivariant Measurable Map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to ``Lorentz breaking'' effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance.
Savini A. - One of the best experts on this subject based on the ideXlab platform.
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A MATSUMOTO–MOSTOW RESULT FOR ZIMMER’S COCYCLES OF HYPERBOLIC LATTICES
'Springer Science and Business Media LLC', 2020Co-Authors: Moraschini M., Savini A.Abstract:Following the philosophy behind the theory of maximal representations, we introduce the volume of a Zimmer's cocycle Gamma x X -> PO degrees (n, 1), where Gamma is a torsion-free (non-)uniform lattice in PO degrees (n, 1), with n > 3, and X is a suitable standard Borel probability Gamma-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 Gamma\(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 degrees (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
Luca Bombelli - One of the best experts on this subject based on the ideXlab platform.
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discreteness without symmetry breaking a theorem
Modern Physics Letters A, 2009Co-Authors: Luca Bombelli, Joe Henson, Rafael D SorkinAbstract:This paper concerns random sprinklings of points into Minkowski spacetime (Poisson processes). It proves that there exists no equivariant Measurable Map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to "Lorentz breaking" effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance.
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discreteness without symmetry breaking a theorem
arXiv: General Relativity and Quantum Cosmology, 2006Co-Authors: Luca Bombelli, Joe Henson, Rafael D SorkinAbstract:This paper concerns sprinklings into Minkowski space (Poisson processes). It proves that there exists no equivariant Measurable Map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to ``Lorentz breaking'' effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance.