Duke Mathematical Journal

The bounded Borel class and 3-manifold groups

Michelle Bucher, Marc Burger, and Alessandra Iozzi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

bounded Borel class rigidity 3-manifold groups complex representations


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

Export citation


  • [1] N. Bergeron, E. Falbel, and A. Guilloux, Tetrahedra of flags, volume and homology of SL(3), Geom. Topol. 18 (2014), no. 4, 1911–1971.
  • [2] N. Bergeron, E. Falbel, A. Guilloux, P.-V. Koseleff, and F. Rouillier, Local rigidity for $\operatorname{PGL}(3,\mathbb{C})$-representations of 3-manifold groups, Exp. Math. 22 (2013), no. 4, 410–420.
  • [3] S. J. Bloch, Higher Regulators, Algebraic $K$-Theory, and Zeta Functions of Elliptic Curves, CRM Monogr. Ser. 11, Amer. Math. Soc., Providence, 2000.
  • [4] R. Brooks, “Some remarks on bounded cohomology” in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (Stony Brook, NY, 1978), Ann. of Math. Stud. 97, Princeton Univ. Press, Princeton, 1981, 53–63.
  • [5] M. Bucher, M. Burger, R. Frigerio, A. Iozzi, C. Pagliantini, and M. B. Pozzetti, Isometric embeddings in bounded cohomology, J. Topol. Anal. 6 (2014), no. 1, 1–25.
  • [6] M. Bucher, M. Burger, and A. Iozzi, “A dual interpretation of the Gromov-Thurston proof of Mostow rigidity and volume rigidity for representations of hyperbolic lattices” in Trends in Harmonic Analysis, Springer INdAM Ser. 3, Springer, Milan, 2013, 47–76.
  • [7] M. Burger and A. Iozzi, Boundary maps in bounded cohomology, appendix to Continuous bounded cohomology and applications to rigidity theory by M. Burger and N. Monod, Geom. Funct. Anal. 12 (2002), no. 2, 281–292.
  • [8] M. Burger, A. Iozzi, and A. Wienhard, Surface group representations with maximal Toledo invariant, Ann. of Math. (2) 172 (2010), no. 1, 517–566.
  • [9] M. Burger and N. Monod, Bounded cohomology of lattices in higher rank Lie groups, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 2, 199–235.
  • [10] M. Burger and N. Monod, “On and around the bounded cohomology of $\mathrm{SL}_{2}$” in Rigidity in Dynamics and Geometry (Cambridge, 2000), Springer, Berlin, 2002, 19–37.
  • [11] J. I. Burgos Gil, The regulators of Beilinson and Borel, CRM Monogr. Ser. 15, Amer. Math. Soc., Providence, 2002.
  • [12] T. Dimofte, M. Gabella, and A. Goncharov, K-decompositions and 3d guage theories, J. High Energy Phys. 2015 no. 11, 151, front matter+144.
  • [13] N. M. Dunfield, Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math. 136 (1999), no. 3, 623–657.
  • [14] E. Falbel and Q. Wang, Duality and invariants of representations of fundamental groups of 3-manifolds into $\operatorname{PGL}(3,\mathbb{C})$, J. Lond. Math. Soc. (2) 95 (2017), no. 1, 1–22.
  • [15] H. Furstenberg, “Boundary theory and stochastic processes on homogeneous spaces” in Harmonic analysis on homogeneous spaces, Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972, Amer. Math. Soc., Providence, R.I., (1973), 193–229.
  • [16] S. Garoufalidis, D. Thurston, and C. Zickert, The complex volume of ${\mathrm{SL} }(n,\mathbb{C})$-representations of 3-manifolds, Duke Math. J. 164 (2015), no. 11, 2099–2160.
  • [17] A.B. Goncharov, “Explicit construction of characteristic classes” in I. M. Gel$'$fand Seminar, Adv. Soviet Math. 16, Amer. Math. Soc., Providence, RI, 1993, 169–210.
  • [18] M. Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99.
  • [19] T. Hartnick and A. Ott, Bounded cohomology via partial differential equations, I, Geom. Topol. 19 (2015), no. 6, 3603–3643.
  • [20] P. Menal-Ferrer and J. Porti, Twisted cohomology for hyperbolic three manifolds, Osaka J. Math. 49 (2012), no. 3, 741–769.
  • [21] N. Monod, “Stabilization for ${\mathrm{SL} }_{n}$ in bounded cohomology” in Discrete Geometric Analysis, Contemp. Math., 347, Amer. Math. Soc., Providence, RI, 2004, 191–202.
  • [22] H. Pieters, The boundary model for the continuous cohomology of $\operatorname{ISOM}^{+}(\mathbb{H}^{n})$, preprint, arXiv:1507.04915 [math.GR].
  • [23] W. Thurston, Geometry and topology of 3-manifolds, notes from Princeton University, Princeton, NJ, 1978.