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

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 tetrahedra complexes. We describe when this volume belongs to the Bloch group. In doing so, we recover and generalize results of Neumann-Zagier, Neumann, and Kabaya. Our approach is very related to the work of Fock and Goncharov.

45 pages, 14 figures. The first version of the paper contained a mistake which is correct here. Hopefully the relation between the works of Neumann-Zagier on one side and Fock-Goncharov on the other side is now much clearer

References in corpus (3)

• A personal account of the discovery of hyperbolic structures on some knot complements
• A combinatorial invariant for Spherical CR structures
• A Volume Function for CR Tetrahedra

Cited by in corpus (17)

• Complex hyperbolic geometry of the figure eight knot
• The complex volume of SL(n,C)-representations of 3-manifolds
• A 1-parameter family of spherical CR uniformizations of the figure eight knot complement
• The bounded Borel class and complex representations of 3-manifold groups
• The symplectic properties of the PGL(n,C)-gluing equations
• Representations of fundamental groups of 3-manifolds into PGL(3,C): Exact computations in low complexity
• Deformation of hyperbolic manifolds in $\mathrm{PGL}(n,\mathbf{C})$ and discreteness of the peripheral representations
• Complex Chern-Simons theory at level k via the 3d-3d correspondence
• A Spectral Perspective on Neumann-Zagier
• Duality and invariants of representations of fundamental groups of 3-manifolds into PGL(3,C)
• Homological and Bloch invariants for Q-rank one spaces and flag structures
• Hyperbolic tessellations and generators of K_3 for imaginary quadratic fields
• Ptolemy coordinates, Dehn invariant, and the A-polynomial
• Volumes of ${\mathrm{SL}}_n\mathbb{C}$-representations of hyperbolic 3-manifolds
• Flag structures on real 3-manifolds
• Tautological characteristic classes I
• Chern-Simons theory and cohomological invariants of representation varieties