 
Summary: IMMERSIONS AND EMBEDDINGS OF TOTALLY
GEODESIC SURFACES
D. D. LONG
The work of Waldhausen, Thurston and others has shown that the existence of
an embedding of a closed, orientable, incompressible surface in a 3manifold is a great
help in the understanding of that manifold. Unfortunately many examples exist of
manifolds which contain no such embedding. However, it does seem at least
conjecturally possible that any irreducible manifold with infinite fundamental group
could contain an immersion of such a surface, and this has motivated the study of
the question of whether such a surface can always be lifted to an embedding in some
finite covering of the 3manifold. The general question seems to be some way from
resolution; the purpose of this note is to give an affirmative answer in a very special
case.
THEOREM 1. Let M be a closed, hyperbolic 3manifold which contains a totally
geodesic immersion of a closed surface. Then:
(a) there is a finite covering of M which contains an embedded closed, orientable,
totally geodesic surface;
(b) there is a finite covering ofM which contains an embedded nonseparating, closed,
orientable, totally geodesic surface.
The restriction to the closed case is unnecessary, and is made only to save some
