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Analytic families of quantum hyperbolic invariants

We organize the quantum hyperbolic invariants (QHI) of $3$-manifolds into sequences of rational functions indexed by the odd integers $N\geq 3$ and defined on moduli spaces of geometric structures refining the character varieties. In the case of one-cusped hyperbolic $3$-manifolds $M$ we generalize the QHI and get rational functions $\mathcal{H}_N^{h_f,h_c,k_c}$ depending on a finite set of cohomological data $(h_f,h_c,k_c)$ called {\it weights}. These functions are regular on a determined Abelian covering of degree $N^2$ of a Zariski open subset, canonically associated to $M$, of the geometric component of the variety of augmented $PSL(2,\mathbb{C})$-characters of $M$. New combinatorial ingredients are a weak version of branchings which exists on every triangulation, and state sums over weakly branched triangulations, including a sign correction which eventually fixes the sign ambiguity of the QHI. We describe in detail the invariants of three cusped manifolds, and present the results of numerical computations showing that the functions $\mathcal{H}_N^{h_f,h_c,k_c}$ depend on the weights as $N\rightarrow + \infty$, and recover the volume for some specific choices of the weights.