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Elementary amenable groups and 4-manifolds with Euler characteristic 0

Part of: Special aspects of infinite or finite groups Topological manifolds Low-dimensional topology

Published online by Cambridge University Press:  09 April 2009

Jonathan A. Hillman
Jonathan A. Hillman
Affiliation:
University of Sydney Sydney, N.S.W. 2006, Australia
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Abstract

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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 asphencal and the group is virtually poly- Z of Hirsch length 4.

Keywords

asphericitydeficiencyelementary amenable groupEuler characteristic4-manifold

MSC classification

Secondary: 57N13: Topology of $E^4$, $4$-manifolds 57M20: Two-dimensional complexes 20F99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 1 , February 1991 , pp. 160 - 170
Copyright
Copyright © Australian Mathematical Society 1991

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