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

Abstract

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 $q=e\frac{2\pi i}{r}$.