The state sums defining the quantum hyperbolic invariants (QHI) of hyperbolic oriented cusped $3$-manifolds can be split in a "symmetrization" factor and a "reduced" state sum. We show that these factors are invariants on their own, that we call "symmetry defects" and "reduced QHI", provided the manifolds are endowed with an additional "non ambiguous structure", a new type of combinatorial structure that we introduce in this paper. A suitably normalized version of the symmetry defects applies to compact $3$-manifolds endowed with $PSL_2(\mathbb{C})$-characters, beyond the case of cusped manifolds. Given a manifold $M$ with non empty boundary, we provide a partial "holographic" description of the non-ambiguous structures in terms of the intrinsic geometric topology of $\partial M$. Special instances of non ambiguous structures can be defined by means of taut triangulations, and the symmetry defects have a particularly nice behaviour on such "taut structures". Natural examples of taut structures are carried by any mapping torus with punctured fibre of negative Euler characteristic, or by sutured manifold hierarchies. For a cusped hyperbolic $3$-manifold $M$ which fibres over $S^1$, we address the question of determining whether the fibrations over a same fibered face of the Thurston ball define the same taut structure. We describe a few examples in detail. In particular, they show that the symmetry defects or the reduced QHI can distinguish taut structures associated to different fibrations of $M$. To support the guess that all this is an instance of a general behaviour of state sum invariants of 3-manifolds based on some theory of 6j-symbols, finally we describe similar results about reduced Turaev-Viro invariants.

中文翻译：

---

三流形上的非二义性结构和量子对称缺陷

定义双曲导向尖头 $3$-流形的量子双曲不变量 (QHI) 的状态总和可以分为“对称化”因子和“约化”状态总和。我们证明这些因素本身是不变量，我们称之为“对称缺陷”和“减少的 QHI”，前提是流形被赋予了额外的“非歧义结构”，这是我们在本文中介绍的一种新型组合结构. 对称缺陷的适当归一化版本适用于具有 $PSL_2(\mathbb{C})$-characters 的紧凑 $3$-流形，超出了尖头流形的情况。给定具有非空边界的流形 $M$，我们根据 $\partial M$ 的内在几何拓扑提供了非歧义结构的部分“全息”描述。非二义性结构的特殊实例可以通过拉紧三角剖分来定义，对称缺陷在这种“拉紧结构”上有特别好的表现。绷紧结构的自然示例由具有负欧拉特征的刺穿纤维的任何映射圆环或由缝合的流形层次结构承载。对于在 $S^1$ 上纤维化的尖头双曲线 $3$-流形 $M$，我们解决了确定 Thurston 球的同一纤维面上的纤维化是否定义相同拉紧结构的问题。我们详细描述了几个例子。特别是，他们表明对称缺陷或降低的 QHI 可以区分与 $M$ 不同纤维化相关的绷紧结构。