We prove the Turaev-Viro invariants volume conjecture for complements of fundamental shadow links: an infinite family of hyperbolic link complements in connected sums of copies of $S^1\times S^2$. The main step of the proof is to find a sharp upper bound on the growth rate of the quantum $6j-$symbol evaluated at $e^{\frac{2\pi i}{r}}.$ As an application of the main result, we show that the volume of any hyperbolic 3-manifold with empty or toroidal boundary can be estimated in terms of the Turaev-Viro invariants of an appropriate link contained in it. We also build additional evidence for a conjecture of Andersen, Masbaum and Ueno (AMU conjecture) about the geometric properties of surface mapping class groups detected by the quantum representations.

中文翻译：

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量子$6j$-符号的增长及其在体积猜想中的应用

我们证明了基本阴影链接补集的 Turaev-Viro 不变量体积猜想：在 $S^1\times S^2$ 的副本的连接和中的无限双曲链接补集。证明的主要步骤是找到在 $e^{\frac{2\pi i}{r}} 处评估的量子 $6j-$symbol 的增长率的急剧上限。主要结果，我们表明任何具有空或环形边界的双曲 3 流形的体积都可以根据其中包含的适当链接的 Turaev-Viro 不变量来估计。我们还为 Andersen、Masbaum 和 Ueno 的猜想（AMU 猜想）建立了关于由量子表示检测到的表面映射类群的几何特性的额外证据。