  We prove the Turaev-Viro invariants volume conjecture for a "universal" class
of cusped hyperbolic 3-manifolds that produces all 3-manifolds with empty or
toroidal boundary by Dehn filling. This leads to two-sided bounds on the volume
of any hyperbolic 3-manifold with empty or toroidal boundary in terms of the
growth rate of the Turaev-Viro invariants of the complement of an appropriate
link contained in the manifold. We also provide evidence for a conjecture of
Andersen, Masbaum and Ueno (AMU conjecture) about certain quantum
representations of surface mapping class groups.
  A key step in our proofs is finding a sharp upper bound on the growth rate of
the quantum $6j-$symbol evaluated at $q=e^{\frac{2\pi i}{r}}.$
