On the Volume Conjectures for Dehn-filled $3$-manifolds along fundamental shadow link complements with big volume

Abstract

We prove the Volume Conjecture and the Asymptotic Expansion Conjecture for closed oriented hyperbolic $3$-manifolds obtained by integral Dehn-fillings along a fundamental shadow link complement, provided that their volumes are sufficiently close to that of the complement. More generally, we show that the conjectured asymptotic expansions hold for pairs obtained by change-of-pair operations on a fundamental shadow link complement whenever the volume of the corresponding hyperbolic cone manifold is sufficiently close to that of the complement. A key ingredient in the proof is the introduction of extended quantum $6j$-symbols and a Weak Reflection Symmetry satisfied by them. The Weak Reflection Symmetry is established by relating the extended quantum $6j$-symbols to the $b$-$6j$ symbols through a real Gevrey-$1$ estimate and a truncated Borel-Laplace transform.