QHI Theory, I: 3-Manifolds Scissors Congruence Classes and Quantum Hyperbolic Invariants

For any triple $(W,L,\rho)$, where W is a closed connected and oriented 3-manifold, L is a link in W and $\rho$ is a flat principal B-bundle over W (B is the Borel subgroup of $SL(2,\mc)$), one constructs a $\Dd$-scissors congruence class $\cG_{\Dd}(W,L,\rho)$ which belongs to a (pre)-Bloch group $\Pp (\Dd)$. The class $\cG_{\Dd}(W,L,\rho)$ may be represented by $\Dd$-triangulations $\Tt=(T,H,\Dd)$ of $(W,L,\rho)$. For any $\Tt$ and any odd integer $N>1$, one defines a ``quantization'' $\Tt_N$ of $\Tt$ based on the representation theory of the quantum Borel subalgebra $\Ww_N$ of $U_q(sl(2,\mc))$ specialized at the root of unity $\omega_N = \exp (2\pi i/N)$. Then one defines an invariant state sum $K_N(W,L,\rho):= K(\Tt_N)$ called a quantum hyperbolic invariant (QHI) of $(W,L,\rho)$. One introduces the class of hyperbolic-like triples. They carry also a classical scissors congruence class $\cG_{\Ii}(W,L,\rho)$, 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 $\cG_{\Ii}(W,L,\rho)$ lies in the kernel of a generalized Dehn homomorphism defined on $\Pp (\Ii)$, and that it induces an element of $H_3^\delta(PSL(2,\mc);\mz)$ (discrete homology). One proves that $\lim_{N\to \infty} (2i\pi/N^2) \log [K_N(W,L,\rho)] = G(W,L,\rho)$ essentially depends of the geometry of the ideal triangulations representing $\cG_{\Ii}(W,L,\rho)$, and one motivates the strong reformulation of the Volume Conjecture, which would identify $G(W,L,\rho)$ with the evaluation $R(\cG_{\Ii}(W,L,\rho))$ of a certain refinement of the classical Rogers dilogarithm on the $\Ii$-scissors class.