Bonahon–Wong–Yang volume conjecture for quantum invariants of surface diffeomorphisms

Let $\varphi:\Sigma\to\Sigma$ be a pseudo-Anosov diffeomorphism, let $M_\varphi$ be its mapping torus, and let $r$ be a generic $\varphi$-invariant character in the smooth part of the $\mathrm{SL}(2;\mathbb C)$-character variety of $\Sigma$. For $\varphi$-invariant puncture weights, let $\Lambda^q_{\varphi,r,p_v}:V\to V$ be the associated intertwiner. Bonahon–Wong–Yang volume conjecture. For the sequence of puncture weights $p_v=e^{h_v/n}+e^{-h_v/n}$,

$$\lim_{\substack{n\to\infty\ \text{ odd}}}\frac{4\pi}{n}\ln\left|\mathrm{Trace}(\Lambda^q_{\varphi,r,p_v})\right|=\mathrm{Vol}(M_\varphi),$$

where $\mathrm{Vol}(M_\varphi)$ is the hyperbolic volume of the mapping torus with its complete hyperbolic structure. This conjecture predicts that the asymptotics of these quantum invariants recover the complete hyperbolic volume; it is proved for the figure-eight-knot complement and, under additional volume assumptions, for some other surface bundles.