Tits alternatives and low dimensional topology.
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.
scientific article; zbMATH DE number 1943315