Volume 18, issue 4 (2014)

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

Nicolas Bergeron, Elisha Falbel and Antonin Guilloux

Geometry & Topology 18 (2014) 1911–1971
Abstract

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].

Keywords
Bloch group, $3$–manifolds, $\mathrm{PGL}(3,\mathbb{C})$, tetrahedra
Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 57N10, 57R20
References
Publication
Received: 30 September 2011
Revised: 3 October 2013
Accepted: 27 February 2014
Published: 2 October 2014
Proposed: Walter Neumann
Seconded: Dmitri Burago, Jean-Pierre Otal
Authors
Nicolas Bergeron
Institut de Mathématiques de Jussieu
Université Pierre et Marie Curie
4 place Jussieu
75252 Paris
France
http://people.math.jussieu.fr/~bergeron
Elisha Falbel
Institut de Mathématiques
Université Pierre et Marie Curie
4 place Jussieu
75252 Paris
France
http://people.math.jussieu.fr/~falbel
Antonin Guilloux
Institut de Mathématiques
Université Pierre et Marie Curie
4 place Jussieu
75252 Paris
France
http://people.math.jussieu.fr/~aguilloux