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IMMERSIONS AND EMBEDDINGS OF TOTALLY GEODESIC SURFACES
 

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 3-manifold 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 3-manifold. 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 3-manifold 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 non-separating, closed,
orientable, totally geodesic surface.
The restriction to the closed case is unnecessary, and is made only to save some

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics