Tetrahedra of flags, volume and homology of SL(3)

Bergeron, Nicolas¹ ; Falbel, Elisha² ; Guilloux, Antonin²

¹ Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris, France
² Institut de Mathématiques, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris, France

Geometry & topology, Tome 18 (2014) no. 4, pp. 1911-1971.

Résumé

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].

Détail

DOI : 10.2140/gt.2014.18.1911

Classification : 57M50, 57N10, 57R20
Keywords: Bloch group, $3$–manifolds, $\mathrm{PGL}(3,\mathbb{C})$, tetrahedra

Bibliographie

[1] N Bergeron, E Falbel, A Guilloux, Local rigidity for $\mathrm{PGL}(3 , \mathbb{C})$ representations of $3$–manifold groups, Exp. Math. 22 (2013) 410

[2] F Bonahon, Shearing hyperbolic surfaces, bending pleated surfaces and Thurston's symplectic form, Ann. Fac. Sci. Toulouse Math. 5 (1996) 233

[3] F Bonahon, A Schläfli-type formula for convex cores of hyperbolic $3$–manifolds, J. Differential Geom. 50 (1998) 25

[4] T Dimofte, M Gabella, A Goncharov, $K\!$–decompositions and $3d$ gauge theories,

[5] E Falbel, A spherical CR structure on the complement of the figure eight knot with discrete holonomy, J. Differential Geom. 79 (2008) 69

[6] E Falbel, A volume function for spherical CR tetrahedra, Q. J. Math. 62 (2011) 397

[7] E Falbel, P V Koseleff, F Rouillier, Representations of fundamental groups of $3$–manifolds into $\mathrm{PGL}(3,\mathbb{C})$: Exact computations in low complexity,

[8] E Falbel, Q Wang, A combinatorial invariant for spherical CR structures, Asian J. Math. 17 (2013) 391

[9] V V Fock, A B Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. (2006) 1

[10] V V Fock, A B Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. 42 (2009) 865

[11] S Garoufalidis, M Goerner, C Zickert, Gluing equations for $\mathrm{PGL}(n,\mathbb{C})$–representations of $3$–manifolds,

[12] S Garoufalidis, D Thurston, C Zickert, The complex volume of $\mathrm{SL}(n,\mathbb{C})$–representations of $3$–manifolds, (2013)

[13] J Genzmer, Sur les triangulations des structures CR–sphériques, PhD thesis, Université Pierre et Marie Curie (2010)

[14] H Jacobowitz, An introduction to CR structures, Mathematical Surveys and Monographs 32, Amer. Math. Soc. (1990)

[15] Y Kabaya, Pre-Bloch invariants of $3$–manifolds with boundary, Topology Appl. 154 (2007) 2656

[16] W D Neumann, Combinatorics of triangulations and the Chern–Simons invariant for hyperbolic $3$–manifolds, from: "Topology '90" (editors B Apanasov, W D Neumann, A W Reid, L Siebenmann), Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter, Berlin (1992) 243

[17] W D Neumann, Extended Bloch group and the Cheeger–Chern–Simons class, Geom. Topol. 8 (2004) 413

[18] W D Neumann, D Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985) 307

[19] R Riley, A personal account of the discovery of hyperbolic structures on some knot complements, Expo. Math. 31 (2013) 104

[20] A A Suslin, Homology of $\mathrm{GL}_{n}$, characteristic classes and Milnor $K\!$–theory, Trudy Mat. Inst. Steklov. 165 (1984) 188

[21] W P Thurston, The geometry and topology of $3$–manifolds, lecture notes, Princeton University (1978–1981)

[22] D Zagier, The dilogarithm function, from: "Frontiers in number theory, physics, and geometry, II" (editors P Cartier, B Julia, P Moussa, P Vanhove), Springer, Berlin (2007) 3

[23] C Zickert, The extended Bloch group and algebraic $K\!$–theory, (2010)