Centralizers of reflections in crystallographic groups

The study of hyperbolic 3-manifolds has recently been recognized as an increasingly important part of 3-manifold theory (see (9)) and for some time the presence of incompressible surfaces in a 3-manifold has been known to be important (see, for example, (4)). A particularly interesting case occurs when the incompressible surface unfolds in the universal covering space into a hyperbolic plane. The fundamental group of the surface is then contained in the stabilizer of the plane, or, what is the same thing, in the centralizer of the reflection defined by the plane. This is one motivation for studying centralizers of reflections in discrete groups of hyperbolic isometries, or, as we shall call them, hyperbolic crystallographic groups .