The deformation space of nonorientable hyperbolic 3–manifolds

We consider nonorientable hyperbolic 3-manifolds of finite volume M 3 .When M 3 has an ideal triangulation , we compute the deformation space of the pair .M 3 ; / (its Neumann-Zagier parameter space).We also determine the variety of representations of 1 .M 3 / in Isom.H 3 / in a neighborhood of the holonomy.As a consequence, when some ends are nonorientable, there are deformations from the variety of representations that cannot be realized as deformations of the pair .M 3 ; /.We also discuss the metric completion of these structures and we illustrate the results on the Gieseking manifold.57K32; 57K35, 57Q99