The deformation space of non-orientable hyperbolic 3-manifolds

We consider non-orientable hyperbolic 3-manifolds of finite volume $M^3$. When $M^3$ has an ideal triangulation $\Delta$, we compute the deformation space of the pair $(M^3, \Delta)$ (its Neumann Zagier parameter space). We also determine the variety of representations of $\pi_1(M^3)$ in $\mathrm{Isom}(\mathbb{H}^3)$ in a neighborhood of the holonomy. As a consequence, when some ends are non-orientable, there are deformations from the variety of representations that cannot be realized as deformations of the pair $(M^3, \Delta)$. We also discuss the metric completion of these structures and we illustrate the results on the Gieseking manifold.