On Margulis cusps of hyperbolic -manifolds

We study the geometry of the Margulis region associated with an irrational screw translation acting on the -dimensional real hyperbolic space. This is an invariant domain with the parabolic fixed point of on its boundary which plays the role of an invariant horoball for a translation in dimensions 3. The boundary of the Margulis region is described in terms of a function which solely depends on the rotation angle of . We obtain an asymptotically universal upper bound for as for arbitrary irrational , as well as lower bounds when is Diophantine and the optimal bound when is of bounded type. We investigate the implications of these results for the geometry of Margulis cusps of hyperbolic -manifolds that correspond to irrational screw translations acting on the universal cover. Among other things, we prove bi-Lipschitz rigidity of these cusps.