Translations and Associated Dirichlet Polyhedra in Complex Hyperbolic Space

We study a nonidentity transvection (i.e. (strictly) hyperbolic isometry) or nonidentity Heisenberg translation f of complex hyperbolic space Hⁿ and a Dirichlet polyhedron P of the cyclic group f. We have four main results: (a) If z & in Hⁿ and the axis of a nonidentity transvection are not complex collinear, then, roughly speaking, any two distinct 'naturally arising' geodesics passing through z are not complex collinear. (b) If g is also a transvection or Heisenberg translation of Hⁿ and z & in Hⁿ such that f(z)=g(z) and f^–1(z)=g^–1(z), then f=g. (c) We classify all this kind of polyhedra up to congruence in Hⁿ. (d) We obtain an equivalent condition for P to be cospinal (which means that the complex spines of the two sides of P coincide) in terms of the distance of the spines of the two sides of P.