Relations in bounded cohomology

Farre, James

doi:10.1112/topo.12150

Cited by 3 publications

(1 citation statement)

References 35 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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Borel and Volume Classes for Dense Representations of Discrete Groups

Farre

2021

International Mathematics Research Notices

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We show that the bounded Borel class of any dense representation $\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $G$. When $n=2$, the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$ is uniformly separated in semi-norm from any other representation $\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$ for which there is a subgroup $H\le G$ on which $\rho $ is still dense but $\rho ^{\prime}$ is discrete or indiscrete but stabilizes a point, line, or plane in ${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of ${\operatorname{PSL}}_2{\mathbb{R}}$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties.

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Gambaudo--Ghys construction on bounded cohomology

Kimura¹

2020

Preprint

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Relations in bounded cohomology

Cited by 3 publications

References 35 publications

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

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

Borel and Volume Classes for Dense Representations of Discrete Groups

Gambaudo--Ghys construction on bounded cohomology

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