Revista Matemática Iberoamericana

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Volume 25, Issue 2, 2009, pp. 739–756
DOI: 10.4171/RMI/581

Published online: 2009-08-31

One-relator groups and proper 3-realizability

Manuel Cárdenas[1], Francisco F. Lasheras[2], Antonio Quintero[3] and Dušan Repovš[4]

(1) Universidad de Sevilla, Spain
(2) Universidad de Sevilla, Spain
(3) Universidad de Sevilla, Spain
(4) University of Ljubljana, Slovenia

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.

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

Cárdenas Manuel, Lasheras Francisco, Quintero Antonio, Repovš Dušan: One-relator groups and proper 3-realizability. Rev. Mat. Iberoamericana 25 (2009), 739-756. doi: 10.4171/RMI/581