On geodesic embeddings of hyperbolic surfaces into hyperbolic 3-manifolds

We consider the problem of when a closed orientable hyperbolic surface admits a totally geodesic embedding into a closed orientable hyperbolic 3-manifold; given a finite isometric group action on the surface, we consider in particular equivariant versions of such an embedding. We prove that an equivariant embedding exists for all finite irreducible group actions on surfaces; such surfaces are known also as quasiplatonic surfaces; in particular, all quasiplatonic surfaces embed geodesically. In the last section, we discuss some cases of more general finite group actions on surfaces.