Duke Mathematical Journal

The bounded Borel class and 3-manifold groups

Michelle Bucher, Marc Burger, and Alessandra Iozzi

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Abstract

If Γ<PSL(2,C) is a lattice, we define an invariant of a representation ΓPSL(n,C) using the Borel class β(n)Hc3(PSL(n,C),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 Γ of the irreducible complex n-dimensional representation of PSL(2,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 Hb3(SL(n,C),R) as a normed space.

Article information

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

Dates
Received: 12 February 2017
Revised: 2 July 2018
First available in Project Euclid: 25 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1540454549

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

Mathematical Reviews number (MathSciNet)
MR3874650

Zentralblatt MATH identifier
07000592

Subjects
Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]
Secondary: 22E41: Continuous cohomology [See also 57R32, 57Txx, 58H10] 57R20: Characteristic classes and numbers 53C24: Rigidity results 57N10: Topology of general 3-manifolds [See also 57Mxx] 57M50: Geometric structures on low-dimensional manifolds

Keywords
bounded Borel class rigidity 3-manifold groups complex representations

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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