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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

Author

Baseilhac, S.; Benedetti, R.

Published

2002

Language

EN