We prove that the Dirichlet fundamental polyhedron for a cyclic group generated by a unipotent or hyperbolic element γ acting on complex hyperbolic n-space centered at an arbitrary point w is bounded by the two hypersurfaces equidistant from the pairs w, γw and w, γ −1 respectively. The proof relies on a convexity property of the distance to an isometric flow containing γ

https://www.ams.org/proc/1992-115-01/S0002-9939-1992-1107276-1/S0002-9939-1992-1107276-1.pdf

Dirichlet polyhedra for cyclic groups in complex hyperbolic space

We give a general introduction to complex hyperbolic geometry and prove that the Dirichlet fundamental polyhedra for cyclic groups of isometries generated by a hyperbolic or unipotent element are two-sided. The faces of Dirichlet polyhedra are formed by equidistant hypersurfaces. In complex hyperbolic space these hypersurfaces are not totally geodesic, and half-spaces are not convex, so the computation of Dirichlet polyhedra is more difficult than in the classical case. The proof for cyclic hyperbolic and unipotent groups relies on a convexity property of the distance to the orbit of an isometric flow to show that a sequence of half-spaces is nested.

Dirichlet polyhedra in complex hyperbolic space

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Dirichlet polyhedra in complex hyperbolic space