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Open 3-manifolds whose fundamental groups have infinite center, and a torus theorem for 3-orbifolds. (English) Zbl 1030.57029
It is well-known that compact orientable irreducible 3-manifolds whose fundamental groups have infinite center result to be Seifert fibered spaces: see the Seifert Fiber Space Theorem [F. Waldhausen, Topology 6, 505-517 (1967; Zbl 0172.48704), P. Scott, Ann. Math. 117, 35-70 (1983; Zbl 0016.57006), P. Tukia, J. Reine Angew. Math. 391, 1-54 (1988; Zbl 0644.30027), D. Gabai, Ann. Math. 136, 447-510 (1992; Zbl 0785.57004), A. Casson and D. Jungreis, Invent. Math. 118, 441-456 (1994; Zbl 0840.57005)]. In the noncompact case, the same general assertion does not hold: see, for instance, [P. Scott and T. Tucker, Q. J. Math., Oxf., II. Ser. 40, 481-499 (1989; Zbl 0692.57006)].
The present strong paper succeeds in characterizing open orientable irreducible Seifert fibered 3-manifolds whose fundamental groups have infinite center in terms of large-scale properties of triangulations. The underlying idea is that triangulations induce a kind of metric structure on manifolds, which allows to define {quasi-isometries} between triangulated 3-manifolds. In this context, the author develops the notions of uniform asphericity, of cyclic homotopy and of M-splitting (i.e., a manifold decomposition by tori and annuli into Seifert fibered pieces of uniformly bounded size), making intensive use of PL minimal surfaces.
As an application of the numerous results obtained, both the Seifert Fiber Space Theorem and the Torus Theorem are generalized to the class \(\mathbf O\) of closed orientable irreducible 3-orbifolds without incompressible turnovers (i.e., spheres with three singular points): (i) if \(\mathcal O \in \mathbf O\) and \(\pi_1(\mathcal O)\) has an infinite cyclic normal subgroup, then \(\mathcal O\) is Seifert fibered; (ii) if \(\mathcal O \in \mathbf O\) and \(\pi_1(\mathcal O)\) has a subgroup isomorphic to \(\mathbb Z^2\), then one of the following holds: \(\mathcal O\) contains an incompressible Euclidean 2-suborbifold or: \(\mathcal O\) is Euclidean or Seifert fibered.

MSC:
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M60 Group actions on manifolds and cell complexes in low dimensions
57M50 General geometric structures on low-dimensional manifolds
57Q99 PL-topology
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