Dirichlet polyhedra for parabolic cyclic groups in complex hyperbolic space
Let \({\mathfrak S} = C^{n - 1} \times R \times R_+\) be the Siegel domain model of complex hyperbolic space equipped with horospherical coordinates. A point \(x = (\zeta,v,u) \in {\mathfrak S}\) corresponds to the point \([2 \zeta,1 - |\zeta |^2 - u + iv\), \(1 + |\zeta |^2 + u - iv]\) in \(PC^{n,1}\). The group of holomorphic isometries is the projective unitary group \(PU (n,1)\). Given a base point \(x_0 \in {\mathfrak S}\) for each \(\gamma \in PU (n,1)\) the equidistant halfspace \({\mathcal H} (\gamma)\) is defined as \({\mathcal H} (\gamma) = \{z \in {\mathfrak S}\); \(\rho (z,z_0) < \rho (z, \gamma \cdot z_0)\}\), where \(\rho\) denotes the canonical metric on \({\mathfrak S}\), and then the Dirichlet fundamental domain for a discrete cyclic subgroup \(G = \langle g \rangle\) of \(PU (n,1)\) is defined by \(D_G (z_0) = \cap_{j \in \mathbb{Z} - \{0\}} {\mathcal H} (g^j)\). The boundary of \(D_G (z_0)\) is a union of faces. Let \(g : {\mathfrak S} \to {\mathfrak S}\) be an ellipto-parabolic isometry such that \(g(2 \zeta,v,u) = (A \zeta, v + 1,u)\), where \(A \in U(n - 1)\). The author proves: (i) If \(g(z_0) = z_0\) then \(D_G (z_0)\) has precisely two faces; (ii) if \(g(z_0) \neq z_0\) then \(D_G (z_0)\) has infinitely many faces.
scientific article; zbMATH DE number 806285