Abstract
We give a generalisation of Shimizu’s lemma to complex or quaternionic hyperbolic space in any dimension for groups of isometries containing an arbitrary parabolic map. This completes a project begun by Kamiya (Hiroshima Math J 13:501–506, 1983). It generalises earlier work of Kamiya, Inkang Kim and Parker. The analogous result for real hyperbolic space is due to Waterman (Adv Math 101:87–113, 1993).
Highlights
1.1 The Context The hyperbolic spaces are HFn, where F is one of the real numbers, the complex numbers, the quaternions or Communicated by Doron Lubinsky.W
In all the other cases, there are more complicated parabolic maps, which are conjugate to Euclidean screw motions
For subgroups of PU(n, 1) containing a general Heisenberg translation, Parker [20,21] gave versions of Shimizu’s lemma both in terms of a bound on the radius of isometric spheres and a precisely invariant horoball or sub-horospherical region.This was generalised to PSp(n, 1) by Kim and Parker [16]
Summary
1.1 The Context The hyperbolic spaces (that is rank 1 symmetric spaces of non-compact type) are HFn , where F is one of the real numbers, the complex numbers, the quaternions or Communicated by Doron Lubinsky. Ohtake gave examples showing that, for n ≥ 4, subgroups of Isom(HRn ) containing a more general parabolic map can have isometric spheres of arbitrarily large radius, or equivalently there can be no precisely invariant horoball [19]. For subgroups of PU(n, 1) containing a general Heisenberg translation, Parker [20,21] gave versions of Shimizu’s lemma both in terms of a bound on the radius of isometric spheres and a precisely invariant horoball or sub-horospherical region.This was generalised to PSp(n, 1) by Kim and Parker [16]. Versions for isometry groups of HC2 containing a loxodromic or elliptic map were given by Basmajian and Miner [1] and Jiang et al [9] These results were extended to HH2 by Kim and Parker [16] and Kim [15]. Markham and Parker [18] obtained a version of Jørgensen’s inequality for the isometry groups of HO2 with certain types of loxodromic map
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