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Tits alternatives and low dimensional topology. (English) Zbl 1056.57002

The author gives results on groups of finite cohomological dimension in certain situations of interest in low dimensional topology. The motivation is Tits’ theorem “every finitely generated linear group is either virtually solvable or contains a nonabelian free subgroup”. The target groups in this paper are groups of cohomological dimension two, \(PD_3\)-groups, the fundamental groups of closed four-manifolds with Euler characteristic zero and two-knot groups. The paper also contains several applications on knot-groups.
One of the main results is that if \(M\) is a closed 4-manifold with Euler characteristic 0 such that \(\pi=\pi(M)\) is locally virtually indicable and has no nonabelian free subgroup, and \(Z[\pi]\) is coherent, then either \(\pi\) is solvable and \(c.d.\pi=2\) or \(\pi\) is virtually \(Z^{2}\) or \(\pi\) has two ends or \(M\) is homeomorphic to an infrasolvmanifold.

MSC:

57M07 Topological methods in group theory
20J05 Homological methods in group theory
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
20F65 Geometric group theory
57M99 General low-dimensional topology
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