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Immersed and virtually embedded \(\pi_1\)-injective surfaces in graph manifolds. (English) Zbl 0979.57007

Let \(M^3\) be a closed 3-manifold; an immersion of a surface \(S\) in \(M^3\) is a virtual embedding if it can be lifted to an embedding of a finite cover of \(S\) in some finite cover of \(M^3\). The present paper takes into account the problem of the existence of immersed (resp. virtually embedded) \(\pi_1\)-injective surfaces of negative Euler characteristic into closed 3-manifolds. As far as graph manifolds \(M^3\) are concerned, the paper yields necessary and sufficient conditions for such immersions (resp. virtual embeddings), which involve the decomposition matrix associated to \(M^3\) (defined in [the author, Topology 36, No. 2, 355-378 (1997; Zbl 0872.57021)]). As a consequence, 3-manifolds are proved to exist, which have immersed \(\pi_1\)-injective surfaces of negative Euler characteristic, with the property that no such surface is virtually embedded.
Note that any infinite surface subgroup of \(\pi_1(M^3)\) comes from a \(\pi_1\)-injective immersion of a surface in \(M^3\); moreover, [P. Scott, J. Lond. Math. Soc., II. Ser. 17, 555-565 (1978; Zbl 0412.57006)] implies that, if the subgroup is separable (i.e. the intersection of the finite index subgroups containing it), then the surface is separable. This fact enables one to translate into purely group-theoretical formulations the main results of the present paper. For example, any link of an isolated complex surface singularity is proved to have no non-abelian surface subgroups.

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M10 Covering spaces and low-dimensional topology
57M05 Fundamental group, presentations, free differential calculus
57R42 Immersions in differential topology
57M07 Topological methods in group theory
57R40 Embeddings in differential topology
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References:

[1] S V Buyalo, V L Kobel’skiĭ, Geometrization of graph-manifolds I: Conformal geometrization, Algebra i Analiz 7 (1995) 1 · Zbl 0855.57010
[2] S V Buyalo, V L Kobel’skiĭ, Geometrization of graph-manifolds II: Isometric geometrization, Algebra i Analiz 7 (1995) 96 · Zbl 0855.57011
[3] S V Buyalo, V L Kobel’skiĭ, Geometrization of infinite graph-manifolds, Algebra i Analiz 8 (1996) 56 · Zbl 0884.57013
[4] S V Buyalo, V L Kobel’skiĭ, Generalized graph-manifolds of nonpositive curvature, Algebra i Analiz 11 (1999) 64 · Zbl 0958.53030
[5] A Jacques, C Lenormand, A Lentin, J F Perrot, Un résultat extrémal en théorie des permutations, C. R. Acad. Sci. Paris Sér. A-B 266 (1968) · Zbl 0153.32601
[6] B Leeb, 3-manifolds with(out) metrics of nonpositive curvature, Invent. Math. 122 (1995) 277 · Zbl 0840.53031 · doi:10.1007/BF01231445
[7] W D Neumann, Commensurability and virtual fibration for graph manifolds, Topology 36 (1997) 355 · Zbl 0872.57021 · doi:10.1016/0040-9383(96)00014-6
[8] W D Neumann, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981) 299 · Zbl 0546.57002 · doi:10.2307/1999331
[9] W D Neumann, Immersed and virtually embedded \(\pi_1\)-injective surfaces in graph manifolds, Algebr. Geom. Topol. 1 (2001) 411 · Zbl 0979.57007 · doi:10.2140/agt.2001.1.411
[10] G A Niblo, D T Wise, The engulfing property for 3-manifolds, Geom. Topol. Monogr. 1, Geom. Topol. Publ., Coventry (1998) 413 · Zbl 0910.57007
[11] G A Niblo, D T Wise, Subgroup separability, knot groups and graph manifolds, Proc. Amer. Math. Soc. 129 (2001) 685 · Zbl 0967.20018 · doi:10.1090/S0002-9939-00-05574-X
[12] J H Rubinstein, S Wang, \(\pi_1\)-injective surfaces in graph manifolds, Comment. Math. Helv. 73 (1998) 499 · Zbl 0916.57001 · doi:10.1007/s000140050066
[13] P Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. \((2)\) 17 (1978) 555 · Zbl 0412.57006 · doi:10.1112/jlms/s2-17.3.555
[14] S Wang, F Yu, Graph manifolds with non-empty boundary are covered by surface bundles, Math. Proc. Cambridge Philos. Soc. 122 (1997) 447 · Zbl 0899.57011 · doi:10.1017/S0305004197001709
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