Uniform discreteness and Heisenberg screw motions (Q1395392)

The purpose of this paper is to give a generalisation of Shimizu's lemma [\textit{H. Shimizu}, Ann. Math. (2) 77, 33-71 (1963; Zbl 0218.10045)], for groups of isometries of complex hyperbolic 2-space \({\mathbf H}_\mathbb C^2\) with a screw-parabolic element \(A\) and so to extend Waterman's result to this case [\textit{P. L. Waterman}, Adv. Math. 101, 87-113 (1993; Zbl 0793.15019)]. In broad outline, the method follows that established by Shimizu. The main idea is that if two elements of a Lie group are close to the identity then their commutator is even closer to the identity. Therefore, given a screw parabolic map \(A\), the authors take a sequence \(B_n\) of elements of \(\langle A, B\rangle\) defined by \(B_0 = B\) and \(B_{n+1} = B_nAB_n^{-l}\). One then finds the entries of \(B_{n+1}\) iteratively in terms of \(B_n\). This iterative system has a fixed point corresponding to the solution \(B_n = A\) for all \(n\). The goal is to find conditions on \(B\) (depending on \(A\)) so that all the \(B_n\) lie in a basin of attraction of this fixed point. Finally, one shows that, under these hypotheses, \(B_n\) tends to \(A\) as \(n\) tends to infinity. Screw motions are the most complicated of all parabolic maps. They combine features of both boundary elliptic maps and Heisenberg translations. Indeed, by letting the rotational part tend to the identity, a screw motion tends to a vertical translation and, similarly, letting the translation length tend to zero, a screw motion tends to a boundary elliptic map. The main result interpolates between known results for groups with a generator that is a vertical translation or a Heisenberg rotation. Specifically, as the rotational part tends to the identity, the result tends to Kamiya's result. On the other hand, as the translation length goes to zero, the result tends toward the version of Jørgensen's inequality for boundary elliptic maps. The main result depends on a normalization of the screw parabolic map as a particular Heisenberg screw motion. A bound on the radii of isometric spheres in terms of the distance of their centres from the axis of the screw-parabolic is given. It is shown that if \(A\) is a screw parabolic map and \(B\) is an element of PU(2,1) with isometric sphere of very large radius \(r_B\) and if \(\langle A, B\rangle \) is discrete, then the centre of the isometric sphere of either \(B\) or \(B^{-1}\) must be very far from the axis of \(A\).