Compactifying coverings of 3-manifolds
Tóm tắt
If a finitely presented groupG is negatively curved, automatic or asynchronously automatic thenG has an asynchronously bounded “almost prefix closed” combing. Results in [Br1] and [E] imply that the fundamental group of any closed 3-manifold satisfying Thurston's geometrization conjecture has an asynchronously bounded, almost prefix closed combing. MAIN THEOREM. IfM is a compactP
2-irreducible 3-manifold,π
1 (M) has an asynchronously bounded, almost prefix closed combing, andH, a subgroup ofπ
1 (M), is quasiconvex with respect to this combing, then the cover ofM corresponding toH is a missing boundary manifold.
Tài liệu tham khảo
[Bo]F. Bonahon,Bouts des variétés hyperboliques de dimension 3, Annals of Math.124 (1986), 71–158.
[Br1]M. Bridson,Combings of semidirect products and 3-manifold groups, Geom. and Funct. Anal.3 No. 3 (1993), 263–278.
[Br2]M. Bridson, Personal communication.
[BH]M. Bridson andA. Haefliger,Metric spaces of non-positive curvature, preprint.
[BT]M. Brin andT. Thickstun,3-manifolds which are end 1-movable, Memoirs of the AMS81 No. 411 (1989).
[E]D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Patterson andW. P. Thurston,Word processing in groups, Jones and Barlett, 1992.
[GS]S. Gersten andH. Short,Rational subgroups of biautomatic groups, Annals of Math.134 (1991), 125–158.
[HRS]J. Hass, H. Rubinstein andP. Scott,Compactifying coverings of closed 3-manifolds, J. Diff. Geom.30 (1989), 817–832.
[J]W. Jaco,Lectures on three-manifold topology, American Math Soc. CBMS Lecture Note Series, 1980.
[L]B. Leeb,3-Manifolds with(out) metrics of non-positive curvature, Invent. Math.122 No. 2 (1995), 277–290.
[MT]M. L. Mihalik andS. T. Tschantz,Tame combings and the quasi-simply-filtered condition for groups, Trans. Amer. Math. Soc., to appear.
[P]V. Poénaru, Almost convex groups, Lipschitz combing, andπ ∞1 for universal convering spaces of closed 3-manifolds, J. Diff. Geom.35 (1992), 103–130.
[R]L. Reeves,Rational subgroups of cubed 3-manifold groups, Michigan Math. J.42 No. 1 (1995), 109–126.
[Si]J. Simon,Compactification of convering spaces of compact 3-manifolds, Mich. Math. J.23 (1976), 245–256.
[T]W. Thurston,Geometry and topology of 3-manifolds, Lecture Notes, Princeton University, Princeton, NJ, 1978–9.
[Tu]T. W. Tucker,Non-compact 3-manifolds and the missing boundary problem, Topology (13) (1974), 267–273.