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.

We propose a volume conjecture for hyperbolic polyhedra that is similar in spirit to the recent volume conjecture by Chen and Yang on the growth of the Turaev-Viro invariants. Using Barrett's Fourier transform we are able to prove this conjecture in a large family of examples. As a consequence of this result, we prove the Turaev-Viro volume conjecture for a new infinite family of hyperbolic manifolds.

We prove the Turaev-Viro invariants volume conjecture for complements of fundamental shadow links: an infinite family of hyperbolic link complements in connected sums of copies of S 1 × S 2 . The main step of the proof is to find a sharp upper bound on the growth rate of the quantum 6j−symbol evaluated at e 2πi r . As an application of the main result, we show that the volume of any hyperbolic 3-manifold with empty or toroidal boundary can be estimated in terms of the Turaev-Viro invariants of an appropriate link contained in it. We also build additional evidence for a conjecture of Andersen, Masbaum and Ueno (AMU conjecture) about the geometric properties of surface mapping class groups detected by the quantum representations.

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We study the supremum of the volume of hyperbolic polyhedra with some fixed combinatorics and with vertices of any kind (real, ideal or hyperideal). We find that the supremum is always equal to the volume of the rectification of the 1-skeleton.The theorem is proved by applying a sort of volume-increasing flow to any hyperbolic polyhedron. Singularities may arise in the flow because some strata of the polyhedron may degenerate to lowerdimensional objects; when this occurs, we need to study carefully the combinatorics of the resulting polyhedron and continue with the flow, until eventually we get a rectified polyhedron. Contents 1 Introduction 2 Generalized hyperbolic polyhedra 3 The space of proper polyhedra and the volume function 3.1 The Bao-Bonahon existence and uniqueness theorem for hyperideal polyhedra .