Gromov norm and Turaev-Viro invariants of 3-manifolds

We establish a relation between the "large r" asymptotics of the Turaev-Viro invariants $TV_r$ and the Gromov norm of 3-manifolds. We show that for any orientable, compact 3-manifold $M$, with (possibly empty) toroidal boundary, $\log |TV_r (M)|$ is bounded above by a function linear in $r$ and whose slope is a positive universal constant times the Gromov norm of $M$. The proof combines TQFT techniques, geometric decomposition theory of 3-manifolds and analytical estimates of $6j$-symbols. We obtain topological criteria that can be used to check whether the growth is actually exponential; that is one has $\log| TV_r (M)|\geqslant B \ r$, for some $B>0$. We use these criteria to construct infinite families of hyperbolic 3-manifolds whose $SO(3)$ Turaev-Viro invariants grow exponentially. These constructions are essential for the results of [DK:AMU] where the authors make progress on a conjecture of Andersen, Masbaum and Ueno about the geometric properties of surface mapping class groups detected by the quantum representations. We also study the behavior of the Turaev-Viro invariants under cutting and gluing of 3-manifolds along tori. In particular, we show that, like the Gromov norm, the values of the invariants do not increase under Dehn filling and we give applications of this result on the question of the extent to which relations between the invariants $TV_r$ and hyperbolic volume are preserved under Dehn filling. Finally we give constructions of 3-manifolds, both with zero and non-zero Gromov norm, for which the Turaev-Viro invariants determine the Gromov norm.

我们建立了图拉耶夫 - 维罗不变量$TV_r$的“大$r$”渐近性与三维流形的格罗莫夫范数之间的关系。我们表明，对于任何具有（可能为空的）环面边界的可定向紧致三维流形$M$，$\log|TV_r(M)|$由一个关于$r$的线性函数从上方界定，其斜率是一个正的普适常数乘以$M$的格罗莫夫范数。证明结合了拓扑量子场论（TQFT）技术、三维流形的几何分解理论以及$6j$ - 符号的分析估计。我们得到了可用于检查增长是否实际上是指数型的拓扑准则；即对于某个$B > 0$，有$\log|TV_r(M)|\geqslant B\ r$。我们使用这些准则构造了双曲三维流形的无穷族，其$SO(3)$图拉耶夫 - 维罗不变量呈指数增长。这些构造对于[DK:AMU]的结果是必不可少的，在其中作者在安徒生、马斯鲍姆和上野关于由量子表示所检测的曲面映射类群的几何性质的一个猜想上取得了进展。我们还研究了图拉耶夫 - 维罗不变量在沿着环面切割和粘贴三维流形时的行为。特别地，我们表明，与格罗莫夫范数一样，在德恩填充下不变量的值不会增加，并且我们给出了这个结果在德恩填充下不变量$TV_r$和双曲体积之间的关系在多大程度上得以保持这一问题上的应用。最后我们给出了格罗莫夫范数为零和非零的三维流形的构造，对于这些流形，图拉耶夫 - 维罗不变量确定了格罗莫夫范数。