On Margulis Cusps of Hyperbolic 4-Manifolds

Viveca Erlandsson (CUNY GC)

Abstract: Consider a discrete subgroup G of the isometry group of hyperbolic n-space and a parabolic fixed point p. The Margulis region consists of all points in the space that are moved a small distance by an isometry in the stabilizer of p in G, and is kept precisely invariant under this stabilizer. In dimensions 2 and 3 the Margulis region is always a horoball, which gives the well-understood picture of the parabolic cusps in the quotient manifold. In higher dimensions, due to the existence of screw-translations (parabolic isometries with a rotational part), this is no longer true. When the screw-translation has an irrational rotation, the shape of the corresponding region depends on the continued fraction expansion of the irrational angle. In this talk we describe the asymptotic shape of the Margulis region in hyperbolic 4-space corresponding to an irrational screw-translation. As a consequence we show that the corresponding parabolic cusps are bi-Lipschitz rigid. This is joint work with Saeed Zakeri.