2002/01/24 by S. Baseilhac, Baseilhac, S., R. Benedetti +1
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #math.GT
paper · pdf · doi:10.48550/arxiv.math/0201240
58 pages, 22 figures
arxiv created 2002/01/24 · arxiv updated 2009/11/30
For any triple (W,L,ρ), where W is a closed connected and oriented 3-manifold, L is a link in W and ρ is a flat principal B-bundle over W (B is the Borel subgroup of SL(2,\mc)), one constructs a \Dd-scissors congruence class \cG\Dd(W,L,ρ) which belongs to a (pre)-Bloch group \Pp (\Dd). The class \cG\Dd(W,L,ρ) may be represented by \Dd-triangulations \Tt=(T,H,\Dd) of (W,L,ρ). For any \Tt and any odd integer N>1, one defines a ``quantization'' \TtN of \Tt based on the representation theory of the quantum Borel subalgebra \WwN of Uq(sl(2,\mc)) specialized at the root of unity ωN = exp (2πi/N). Then one defines an invariant state sum KN(W,L,ρ):= K(\TtN) called a quantum hyperbolic invariant (QHI) of (W,L,ρ). One introduces the class of hyperbolic-like triples. They carry also a classical scissors congruence class \cG\Ii(W,L,ρ), that belongs to the classical (pre)-Bloch group \Pp (\Ii) and may be represented by explicit idealizations \Tt\Ii of some \Dd-triangulations \Tt of a special type. One shows that \cG\Ii(W,L,ρ) lies in the kernel of a generalized Dehn homomorphism defined on \Pp (\Ii), and that it induces an element of H3δ(PSL(2,\mc);\mz) (discrete homology). One proves that limN→ ∞ (2iπ/N2) log [KN(W,L,ρ)] = G(W,L,ρ) essentially depends of the geometry of the ideal triangulations representing \cG\Ii(W,L,ρ), and one motivates the strong reformulation of the Volume Conjecture, which would identify G(W,L,ρ) with the evaluation R(\cG\Ii(W,L,ρ)) of a certain refinement of the classical Rogers dilogarithm on the \Ii-scissors class.