J Austral Math Soc Ser A 50 pp160--170, 1991.

Elementary Amenable Groups and 4-Manifolds with Euler Characteristic 0

Jonathan A. Hillman

Abstract

We extend earlier work relating asphericity and Euler characteristic 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 ot 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.

1980 AMS Subject Classification (1985 Revision): primary 57N13; secondary 57M20, 20F99

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Authors

Jonathan A. Hillman

University of Sydney, Sydney, N.S.W. 2006, Australia.