How different is the universal cover of a given finite -complex from a -manifold (from the proper homotopy viewpoint)? Regarding this question, we recall that a finitely presented group is said to be properly -realizable if there exists a compact -polyhedron with whose universal cover has the proper homotopy type of a PL -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 -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 D. Repovš, One-relator groups and proper 3-realizability. Rev. Mat. Iberoam. 25 (2009), no. 2, pp. 739–756DOI 10.4171/RMI/581