Asymptotics of quantum $6j$-symbols and generalized hyperbolic tetrahedra

We establish the geometry behind the quantum $6j$-symbols under only the admissibility conditions as in the definition of the Turaev-Viro invariants of $3$-manifolds. As a classification, we show that the $6$-tuples in the quantum $6j$-symbols give in a precise way to the dihedral angles of (1) a spherical tetrahedron, (2) a generalized Euclidean tetrahedron, (3) a generalized hyperbolic tetrahedron or (4) in the degenerate case the angles between four oriented straight lines in the Euclidean plane. We also show that for a large proportion of the cases, the $6$-tuples always give the dihedral angles of a generalized hyperbolic tetrahedron and the exponential growth rate of the corresponding quantum $6j$-symbols equals the suitably defined volume of this generalized hyperbolic tetrahedron. It is worth mentioning that the volume of a generalized hyperbolic tetrahedron can be negative, hence the corresponding sequence of the quantum $6j$-symbols could decay exponentially. This is a phenomenon that has never been aware of before.