On Margulis cusps of hyperbolic 4-manifolds
We study the geometry of the Margulis region associated with an irrational screw translation $g$ acting on the 4-dimensional real hyperbolic space. This is an invariant domain with the parabolic fixed point of $g$ on its boundary which plays the role of an invariant horoball for a translation in dimensions $\leq 3$. The boundary of the Margulis region is described in terms of a function $B_α: [0,\infty) \to {\mathbb R}$ which solely depends on the rotation angle $α\in {\mathbb R}/{\mathbb Z}$ of $g$. We obtain an asymptotically universal upper bound for $B_α(r)$ as $r \to \infty$ for arbitrary irrational $α$, as well as lower bounds when $α$ is Diophatine and the optimal bound when $α$ is of bounded type. We investigate the implications of these results for the geometry of Margulis cusps of hyperbolic 4-manifolds that correspond to irrational screw translations acting on the universal cover. Among other things, we prove bi-Lipschitz rigidity of these cusps.