One-relator groups and proper 3-realizability

Details One-relator groups and proper 3-realizability One-relator groups and proper 3-realizability

2009-10-02

arxiv.org/abs/0910.0305v2

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

Mathematics

Geometric Topology

Scientific paper

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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