Asymptotics of quantum $6j$ symbols

Abstract

Asymptotics of quantum $6j$ symbols corresponding to a hyperbolic tetrahedra is investigated and the first two leading terms are determined for the case that the tetrahedron has a ideal or ultra-ideal vertex. These terms are given by the volume and the determinant of the Gram matrix of the tetrahedron. A relation to the volume conjecture of the Turaev-Viro invariant is also discussed.