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 $SL(2,\text{mc})$), one constructs a $\text{Dd}$-scissors congruence class ${\text{cG}}_{\text{Dd}}(W,L,\rho )$ which belongs to a (pre)-Bloch group $\text{Pp}(\text{Dd})$. The class ${\text{cG}}_{\text{Dd}}(W,L,\rho )$ may be represented by $\text{Dd}$-triangulations $\text{Tt}=(T,H,\text{Dd})$ of $(W,L,\rho )$. For any $\text{Tt}$ and any odd integer $N>1$, one defines a ``quantization’’ ${\text{Tt}}_N$ of $\text{Tt}$ based on the representation theory of the quantum Borel subalgebra ${\text{Ww}}_N$ of $U_q(sl(2,\text{mc}))$ specialized at the root of unity ${\omega }_N=\exp (2\pi i\mathrm{/}N)$. Then one defines an invariant state sum $K_N(W,L,\rho ):=K({\text{Tt}}_N)$ 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 ${\text{cG}}_{\text{Ii}}(W,L,\rho )$, that belongs to the classical (pre)-Bloch group $\Pp (\Ii)$andmayberepresentedbyexplicitidealizations$\Tt_{\Ii}$ofsome$\Dd$-triangulations$\Tt$ofaspecialtype\mathrm{.}Oneshowsthat$\cG_{\Ii}(W,L,\rho)$ lies in the kernel of a generalized Dehn homomorphism defined on $\text{Pp}(\text{Ii})$, and that it induces an element of $H_3^{\delta }(PSL(2,\text{mc});\text{mz})$ (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 ${\text{cG}}_{\text{Ii}}(W,L,\rho )$, and one motivates the strong reformulation of the Volume Conjecture, which would identify $G(W,L,\rho )$ with the evaluation $R({\text{cG}}_{\text{Ii}}(W,L,\rho ))$ of a certain refinement of the classical Rogers dilogarithm on the $\text{Ii}$-scissors class.