Let $H$ be a simply-connected, complete space of constant negative curvature, that is, a hyperbolic space, $\Gamma$ a discrete group of isometries, and and $M=H/\Gamma$ its quotient space; we assume that $M$ has finite volume.
It is known, I think, that $M$ is "of finite geometric type", that is, there exists a fundamental domain $D\subset H$ which is a finite hyperbolic polyhedron.
There is a universal way of building such polyhedra: we take a point $x\in H$, and consider its $\Gamma$-orbit $\Gamma\cdot x$. Define the Dirichlet-Voronoi polyhedron $P$ of $\Gamma\cdot x$ as the set of all $y\in H$ such that the distance from $y$ to $x$ is less than the distance to any other point of $\Gamma\cdot x$.
It is clear that $P$ is a fundamental domain.
Is it true that $P$ is a finite polyhedron for all $x\in H$? I think I can find this statement in the literature for dimension 3. Somebody told me this can be false in dimension more than 3.
A proof (or disproof) of this statement or a reference to it would be highly appreciated.