One-relator groups and proper 3-realizability

  • Manuel Cárdenas

    Universidad de Sevilla, Spain
  • Francisco F. Lasheras

    Universidad de Sevilla, Spain
  • Antonio Quintero

    Universidad de Sevilla, Spain
  • Dušan D. Repovš

    University of Ljubljana, Slovenia


How different is the universal cover of a given finite 22-complex from a 33-manifold (from the proper homotopy viewpoint)? Regarding this question, we recall that a finitely presented group GG is said to be properly 33-realizable if there exists a compact 22-polyhedron KK with π1(K)G\pi_1(K) \cong G whose universal cover K~\tilde{K} has the proper homotopy type of a PL 33-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 33-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.

Cite this article

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. 739–756

DOI 10.4171/RMI/581