# 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…

## 3 Citations

### The Volume of complete anti-de Sitter 3-manifolds

- Mathematics
- 2015

Up to a finite cover, closed anti-de Sitter $3$-manifolds are quotients of $\mathrm{SO}_0(2,1)$ by a discrete subgroup of $\mathrm{SO}_0(2,1) \times \mathrm{SO}_0(2,1)$ of the form \[j\times…

### On deformation spaces of nonuniform hyperbolic lattices

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2016

Abstract Let Γ be a nonuniform lattice acting on the real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected…

### A note on the integrality of volumes of representations

- Mathematics
- 2022

Let Γ be a torsion-free, non-uniform lattice in SO ( 2 n, 1 ) . We present an elementary, combinatorial–geometrical proof of a theorem of Bucher, Burger, and Iozzi in [BBI21] which states that the…

## References

SHOWING 1-10 OF 44 REFERENCES

### Bounded differential forms, generalized Milnor–Wood inequality and an application to deformation rigidity

- Mathematics
- 2007

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

- Mathematics
- 2013

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

- Mathematics
- 2004

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

- Mathematics
- 2003

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

- Mathematics
- 2007

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

- Mathematics
- 1982

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

- Mathematics
- 2012

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

- Mathematics
- 1999

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

- Mathematics
- 1994

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…