Revista Matemática Iberoamericana

One-relator groups and proper $3$-realizability

Manuel Cárdenas , Francisco F. Lasheras , Antonio Quintero , and Dušan Repovš

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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.

Article information

Rev. Mat. Iberoamericana, Volume 25, Number 2 (2009), 739-756.

First available in Project Euclid: 13 October 2009

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Zentralblatt MATH identifier

Primary: 57M07: Topological methods in group theory
Secondary: 57M10: Covering spaces 57M20: Two-dimensional complexes

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


Cárdenas, Manuel; Lasheras, Francisco F.; Quintero, Antonio; Repovš, Dušan. One-relator groups and proper $3$-realizability. Rev. Mat. Iberoamericana 25 (2009), no. 2, 739--756.

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