It is known that there are
nine compact tetrahedra in three-dimensional hyperbolic space for which the group of
isometries generated by reflections in the faces is discrete. The subgroup of
orientation-perserving isometries in such a group will be called a tetrahedral Kleinian
group. Exactly eight such Γ are arithmetic and also a complete list of the finitely
many arithmetic Fuchsian triangle groups G is known. In this paper, we determine
for which pairs of groups (G,Γ) as above, with one possible exception, one can
embed G into Γ. We find that there are many such pairs, contrasting with the
single pair (G,Γ) which is known to arise when, instead of arithmeticity,
the condition that G be realised as the subgroup of elements of Γ which
centralise a reflection in one of the faces of the associated tetrahedron, is
imposed.
