The Margulis region and screw parabolic elements of bounded type (Q2865915)

By Margulis' lemma, there exists a positive constant \(\epsilon\) depending only on \(n\) such that for any discrete subgroup \(G\) of the group of isometries of the hyperbolic \(n\)-space \(\mathbf{H}^n\), the set of points of \(\mathbf{H}^n\) translated by some element of infinite order in \(G\) by a distance less than \(\epsilon\) is a disjoint union of tubular neighbourhoods of geodesic axes and neighbourhoods of parabolic fixed points, called Margulis regions, both of which are precisely invariant under \(G\). In dimensions \(2\) and \(3\), the Margulis regions can be taken to be horoballs, but in higher dimensions this is not the case if they are stabilised by screw parabolic elements. Based on the work of Susskind, the author analyses a function describing the geometry of the boundary of a Margulis region. As a corollary, the author shows that the Margulis region stabilised by a screw parabolic element whose screw angle is irrational and is bounded with respect to the continued fractional expansion is quasi-isometric to a horoball.