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If G is an elementary amenable group of finite Hirsch length h , then the quotient of G by its maximal locally finite normal subgroup has a maximal solvable normal subgroup, of derived length and index bounded in terms of h .

We determine strong constraints on the generalized Euler invariants of Seifert bundles over non‐orientable base orbifolds which may embed as topologically locally flat submanifolds of S4. In particular, a circle bundle over a non‐orientable surface F embeds if and only if it embeds as the boundary of a regular neighbourhood of an embedding of F in S4, and we show that precisely thirteen geometric 3‐manifolds with elementary amenable fundamental groups embed. With the exception of the Poincaré homology sphere, each member of the latter class may be obtained by 0‐framed surgery on a link which is the union of two slice links, and so embeds smoothly in S4. 1991 Mathematics Subject Classification: 57N13.

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We extend earlier work relating asphericity and Euler characteristics for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups. In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical. Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends or is virtually an extension of Z by a subgroup of Q, or the manifold is aspherical and the group is virtually poly-Z of Hirsch length 4.

Murasugi discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including to knots with trivial Alexander polynomial, such as the two polynomial 1 knots with 11 crossings.Hartley found a restrictive condition satisfied by the Alexander polynomial of any freely periodic knot. We generalize this result to the twisted Alexander polynomial and illustrate the applicability of this extension in cases in which Hartley's criterion does not apply. 57M25, 57M27