Geometry & Topology

Tetrahedra of flags, volume and homology of $\mathrm{SL}(3)$

Nicolas Bergeron, Elisha Falbel, and Antonin Guilloux

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In the paper we define a “volume” for simplicial complexes of flag tetrahedra. This generalizes and unifies the classical volume of hyperbolic manifolds and the volume of CR tetrahedral complexes considered in Falbel [Q. J. Math. 62 (2011) 397–415], and Falbel and Wang [Asian J. Math. 17 (2013) 391–422]. We describe when this volume belongs to the Bloch group and more generally describe a variation formula in terms of boundary data. In doing so, we recover and generalize results of Neumann and Zagier [Topology 24 (1985) 307–332], Neumann [Topology ’90 (1992) 243–271] and Kabaya [Topology Appl. 154 (2007) 2656–2671]. Our approach is very related to the work of Fock and Goncharov [Publ. Math. Inst. Hautes Études Sci. 103 (2006) 1–211; Ann. Sci. Éc. Norm. Supér. 42 (2009) 865–930].

Article information

Geom. Topol., Volume 18, Number 4 (2014), 1911-1971.

Received: 30 September 2011
Revised: 3 October 2013
Accepted: 27 February 2014
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57N10: Topology of general 3-manifolds [See also 57Mxx] 57R20: Characteristic classes and numbers

Bloch group $3$–manifolds $\mathrm{PGL}(3,\mathbb{C})$ tetrahedra


Bergeron, Nicolas; Falbel, Elisha; Guilloux, Antonin. Tetrahedra of flags, volume and homology of $\mathrm{SL}(3)$. Geom. Topol. 18 (2014), no. 4, 1911--1971. doi:10.2140/gt.2014.18.1911.

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