Discrete subgroups of complex hyperbolic motions

Shimizu’s Lemma for Quaternionic Hyperbolic Space

“…Our first main result is the following theorem relating the radius of the isometric spheres of S and S −1 , the Cygan translation length of T at their centres and the Euclidean translation length of the vertical projection of T at the vertical projections of the centres. Theorem 1.1 Let be a discrete subgroup of PSp(n, 1) containing the parabolic map T given by (1). Let : N 4n−1 −→ H n−1 be vertical projection given by : (ζ, v) −→ ζ .…”

“…Suppose that G ∞ the stabiliser of ∞ in G is a cyclic group generated by a parabolic map of the form (1). Suppose that N U,μ and N μ defined by (3) and (4) satisfy N μ < 1/4 and N U,μ < (3 − 2 2 + N μ )/2 and let K be given by (5).…”

“…That is, S ∈ Sp(n, 1) if and only if S(z), S(w) = z, w for all z and w in H n, 1 . From this we find S −1 = H −1 S * H .…”

“…Jørgensen's inequality has also been generalised to other hyperbolic spaces. Versions for isometry groups of H 2 C containing a loxodromic or elliptic map were given by Basmajian and Miner [1] and Jiang et al [9]. These results were extended to H 2 H by Kim and Parker [16] and Kim [15].…”

“…Versions of Shimizu's lemma for subgroups of PU(2, 1) containing a screw parabolic map were given by Jiang et al [10,14]. Kim claimed the main result of [10] holds for PSp (2,1) [15]. But in fact, he failed to consider all possible types of screw parabolic map (in the language below, he assumed μ = 1).…”