Growth of quantum $6j$-symbols and applications to the volume conjecture

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 $S^1\times S^2$. The main step of the proof is to find a sharp upper bound on the growth rate of the quantum $6j-$symbol evaluated at $e^{\frac{2\pi i}{r}}.$ 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.