Growth of quantum 6j-symbols and applications to the Volume Conjecture

TL;DR

This paper proves the volume conjecture for Turaev-Viro invariants in a broad class of hyperbolic 3-manifolds, establishing bounds on their volume via quantum invariants and providing evidence for related conjectures in quantum topology.

Contribution

It introduces a universal class of manifolds for which the volume conjecture holds and derives bounds on hyperbolic volume using quantum invariants, advancing understanding of quantum topology.

Findings

Proved the Turaev-Viro volume conjecture for a broad class of hyperbolic 3-manifolds.

Established bounds on hyperbolic volume based on quantum invariants.

Provided evidence supporting the AMU conjecture on quantum representations.

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