## Duke Mathematical Journal

### The bounded Borel class and $3$-manifold groups

#### Abstract

If $\Gamma\lt \operatorname{PSL}(2,\mathbb{C})$ is a lattice, we define an invariant of a representation $\Gamma\rightarrow\operatorname{PSL}(n,\allowbreak\mathbb{C})$ using the Borel class $\beta(n)\in\mathrm{H}_{\mathrm{c}}^{3}(\operatorname{PSL}(n,\mathbb{C}),\mathbb{R})$. We show that this invariant satisfies a Milnor–Wood type inequality and its maximal value is attained precisely by the representations conjugate to the restriction to $\Gamma$ of the irreducible complex $n$-dimensional representation of $\operatorname{PSL}(2,\mathbb{C})$ or its complex conjugate. Major ingredients of independent interest are the study of our extension to degenerate configurations of flags of a cocycle defined by Goncharov, as well as the identification of $\mathrm{H}_{\mathrm{b}}^{3}(\operatorname{SL}(n,\mathbb{C}),\mathbb{R})$ as a normed space.

#### Article information

Source
Duke Math. J., Volume 167, Number 17 (2018), 3129-3169.

Dates
Revised: 2 July 2018
First available in Project Euclid: 25 October 2018

https://projecteuclid.org/euclid.dmj/1540454549

Digital Object Identifier
doi:10.1215/00127094-2018-0038

Mathematical Reviews number (MathSciNet)
MR3874650

Zentralblatt MATH identifier
07000592

#### Citation

Bucher, Michelle; Burger, Marc; Iozzi, Alessandra. The bounded Borel class and $3$ -manifold groups. Duke Math. J. 167 (2018), no. 17, 3129--3169. doi:10.1215/00127094-2018-0038. https://projecteuclid.org/euclid.dmj/1540454549

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