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Mediterranean Journal of Mathematics

, Volume 13, Issue 5, pp 3303–3309 | Cite as

A Note on Group Extensions and Proper 3-Realizability

  • M. Cárdenas
  • F. F. LasherasEmail author
  • A. Quintero
  • R. Roy
Article

Abstract

The interaction between the study of three-dimensional manifolds and a particular stream of group theory has often been fruitful. In the realm of this, we recall that a finitely presented group G is properly 3-realizable if for some finite 2-dimensional CW-complex X with \({\pi_1(X) \cong G}\), the universal cover of X has the proper homotopy type of a 3-manifold. In this paper, we generalize a previous result on the direct products of groups; more precisely, we show that if \({N \rightarrow G \rightarrow Q}\) is a short exact sequence of infinite finitely presented groups, then G is properly 3-realizable. In particular, any semidirect product of two infinite finitely presented groups is properly 3-realizable. As an application, we show proper 3-realizability for certain classes of groups.

Mathematics Subject Classification

Primary 57M07 Secondary 57M10 57M20 

Keywords

proper homotopy fundamental pro-group properly 3-realizable 3-manifold extensions of groups semidirect products 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • M. Cárdenas
    • 1
  • F. F. Lasheras
    • 1
    Email author
  • A. Quintero
    • 1
  • R. Roy
    • 2
  1. 1.Departamento de Geometría y TopologíaUniversidad de SevillaSevillaSpain
  2. 2.New York Institute of TechnologyOld WestburyUSA

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