COMPACTIFYING COVERINGS OF 3-MANIFOLDS

If a finitely presented group G is negatively curved, automatic or asy nchronously automatic then G has an asynchronously bounded ''almost pr efix closed'' combing. Results in [Br-1] and [E] imply that the fundam ental group of any closed 3-manifold satisfying Thurston's geometrizat ion conjecture has an asynchronously bounded, almost prefix closed com bing. MAIN THEOREM. If M is a compact P-2-irreducible 3-manifold, pi(1 )(M) has an asynchronously bounded, almost prefix closed combing, and H, a subgroup of pi(1)(M), is quasiconvex with respect to this combing , then the cover of hi corresponding to I-I is a missing boundary mani fold.