Mirror symmetries of hyperbolic tetrahedral manifolds

Let Λ be the group generated by reflections in faces of a Coxeter tetrahedron in the hyperbolic space ³: A tetrahedral manifold is a hyperbolic manifold M= ³=Γ uniformized by a torsion free subgroup Γ of the group Λ: By a mirror symmetry we mean an orientation reversing isometry of the manifold acting by reflection. The aim of the paper to investigate mirror symmetries of the tetrahedral manifolds.