Asymptotic behavior of quantum invariants

In this thesis we address the problem of the rate of growth of quantum invariants, specifically the Turaev-Viro invariants of compact manifolds and the related Yokota invariants for embedded graphs. We prove the recent volume conjecture proposed by Chen and Yang in two interesting families of hyperbolic manifolds. Furthermore we propose a similar conjecture for the growth of a certain quantum invariant of planar graphs, and prove it in a large family of examples. This new conjecture naturally leads to the problem of finding the supremum of the volume function among all proper hyperbolic polyhedra with a fixed 1-skeleton; we prove that the supremum is always achieved at the rectification of the 1-skeleton.