Integrality of volumes of representations

@article{Bucher2021IntegralityOV,
  title={Integrality of volumes of representations},
  author={Michelle Bucher and M. Burger and A. Iozzi},
  journal={Mathematische Annalen},
  year={2021}
}
<jats:p>Let <jats:italic>M</jats:italic> be an oriented complete hyperbolic <jats:italic>n</jats:italic>-manifold of finite volume. Using the definition of volume of a representation previously given by the authors in [3] we show that the volume of a representation <jats:inline-formula><jats:alternatives><jats:tex-math>$$\rho :\pi _1(M)\rightarrow \mathrm {Isom}^+({{\mathbb {H}}}^n)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow… 
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References

SHOWING 1-10 OF 52 REFERENCES
Bounded differential forms, generalized Milnor–Wood inequality and an application to deformation rigidity
We establish sufficient conditions for a cohomology class of a discrete subgroup Γ of a connected semisimple Lie group with finite center to be representable by a bounded differential form on the
A Dual Interpretation of the Gromov–Thurston Proof of Mostow Rigidity and Volume Rigidity for Representations of Hyperbolic Lattices
We use bounded cohomology to define a notion of volume of an \(\operatorname{SO}(n,1)\)-valued representation of a lattice \(\varGamma<\operatorname{SO}(n,1)\) and, using this tool, we give a
Maximal Volume Representations are Fuchsian
AbstractWe prove a volume-rigidity theorem for Fuchsian representations of fundamental groups of hyperbolic k-manifolds into Isom $$\mathbb{H}^n$$. Namely, we show that if M is a complete hyperbolic
Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds
Let W be a compact manifold and let ρ be a representation of its fundamental group into PSL(2, C). Then the volume of ρ is defined by taking any ρ-equivariant map from the universal cover W~ to ℍ 3
Inégalités de Milnor-Wood géométriques
EnglishWe prove a generalisation of the celebrated Milnor?Wood inequality. If Y is a closed Riemannian manifold, we consider a representation of its fundamental group into the isometry group of a
Characteristic classes and representations of discrete subgroups of Lie groups
A volume invariant is used to characterize those representations of a countable group into a connected semisimple Lie group G which are injective and whose image is a discrete cocompact subgroup of
The norm of the Euler class
We prove that the norm of the Euler class $${\mathcal {E}}$$ for flat vector bundles is 2−n (in even dimension n, since it vanishes in odd dimension). This shows that the Sullivan–Smillie bound
Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds
This paper proves a theorem about Dehn surgery using a new theorem about PSL2C character varieties. Confirming a conjecture of Boyer and Zhang, this p aper shows that a small hyperbolic knot in a
Rationality of secondary classes
We prove the Bloch conjecture : $ c_2(E) \in H^4_\cald (X,\bbz(2))$ is torsion for holomorphic rank two vector bundles $E$ with an integrable connection over a complex projective variety $X$. We
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