Dirichlet polyhedra for dihedral groups acting on complex hyperbolic space (Q1203501)

We consider groups \(\Gamma\) generated by inversions in a pair of asymptotic complex hyperplanes in complex hyperbolic space. We show that there exists a \(\Gamma\)-invariant real hypersurface \(F\) such that the Dirichlet fundamental polyhedron for \(\Gamma\) centred at \(z_ 0\) has to sides (respectively infinitely many sides) if and only if \(z_ 0\in F\) (respectively \(z_ 0\notin F)\). The Dirichlet regions are determined explicitly in terms of coordinates on \(\Gamma\)-invariant horospheres and the geometry of complex hyperbolic space is developed in terms of these horospherical coordinates.