The Margulis region and screw parabolic elements of bounded type

TL;DR

This paper analyzes the shape and asymptotic behavior of Margulis regions in hyperbolic space influenced by screw parabolic elements of bounded type, showing they are quasi-isometric to horoballs.

Contribution

It describes the asymptotic boundary behavior of Margulis regions with bounded type screw parabolics and proves their quasi-isometry to horoballs in higher-dimensional hyperbolic space.

Findings

Margulis regions with bounded type screw parabolics are quasi-isometric to horoballs.

The boundary of these regions exhibits specific asymptotic behavior.

Quasi-isometry class of Margulis regions is independent of certain screw parabolics.

Abstract

Given a discrete subgroup of the isometries of n-dimensional hyperbolic space there is always a region kept precisely invariant under the stabilizer of a parabolic fixed point, called the Margulis region. While in dimensions 2 and 3 this region is a horoball, it has in general a more complicated shape due to the existence of screw parabolic elements in higher dimensions. In fact, P. Susskind has shown that in a discrete group acting on hyperbolic 4-space containing a screw parabolic element with irrational rotation, the corresponding Margulis region does not contain a horoball. In this paper we describe the asymptotic behavior of the boundary of the Margulis region when the irrational screw parabolic is of bounded type. As a corollary we show that the region is quasi-isometric to a horoball. Although Y. Kim has shown that two screw parabolic isometries with irrational rotation are not conjugate by any quasi-isometry of hyperbolic 4-space, this corollary implies that their corresponding Margulis regions (in the bounded type case) are quasi-isometric.