Open Access
June, 2009 One-relator groups and proper $3$-realizability
Manuel Cárdenas , Francisco F. Lasheras , Antonio Quintero , Dušan Repovš
Rev. Mat. Iberoamericana 25(2): 739-756 (June, 2009).


How different is the universal cover of a given finite $2$-complex from a $3$-manifold (from the proper homotopy viewpoint)? Regarding this question, we recall that a finitely presented group $G$ is said to be properly $3$-realizable if there exists a compact $2$-polyhedron $K$ with $\pi_1(K) \cong G$ whose universal cover $\tilde{K}$ has the proper homotopy type of a PL $3$-manifold (with boundary). In this paper, we study the asymptotic behavior of finitely generated one-relator groups and show that those having finitely many ends are properly $3$-realizable, by describing what the fundamental pro-group looks like, showing a property of one-relator groups which is stronger than the QSF property of Brick (from the proper homotopy viewpoint) and giving an alternative proof of the fact that one-relator groups are semistable at infinity.


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Manuel Cárdenas . Francisco F. Lasheras . Antonio Quintero . Dušan Repovš . "One-relator groups and proper $3$-realizability." Rev. Mat. Iberoamericana 25 (2) 739 - 756, June, 2009.


Published: June, 2009
First available in Project Euclid: 13 October 2009

zbMATH: 1182.57002
MathSciNet: MR2569552

Primary: 57M07
Secondary: 57M10 , 57M20

Keywords: end of group , one-relator group , polyhedron , proper $3$-realizability , proper homotopy equivalence

Rights: Copyright © 2009 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.25 • No. 2 • June, 2009
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