Revista Matemática Iberoamericana

One-relator groups and proper $3$-realizability

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

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Abstract

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

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

Dates
First available in Project Euclid: 13 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1255440073

Mathematical Reviews number (MathSciNet)
MR2569552

Zentralblatt MATH identifier
1182.57002

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

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

Citation

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. https://projecteuclid.org/euclid.rmi/1255440073


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References

  • Ayala, R., Cárdenas, M., Lasheras, F.F. and Quintero, A.: Properly $3$-realizable groups. Proc. Amer. Math. Soc. 133 (2005), no. 5, 1527-1535.
  • Brick, S.G. and Mihalik, M.: The QSF property for groups and spaces. Math. Z. 220 (1995), 207-217.
  • Brown, R. and Huebschmann, J.: Identities among relations. In Low-dimensional topology (Bangor, 1979), 153-202. London Math. Soc. Lecture Note Ser. 48. Cambridge Univ. Press, Cambridge-New York, 1982.
  • Cárdenas, M., Fernández, T., Lasheras, F.F. and Quintero, A.: Embedding proper homotopy types. Colloq. Math. 95 (2003), no. 1, 1-20.
  • Cárdenas, M. and Lasheras, F.F.: Properly $3$-realizable groups: a survey. In Geometric methods in group theory, 1-9. Contemp. Math. 372. Amer. Math. Soc., Providence, RI, 2005.
  • Cárdenas, M., Lasheras, F.F. and Quintero, A.: Proper homotopy invariants of properly $3$-realizable groups. Preprint.
  • Cárdenas, M., Lasheras, F.F., Quintero, A. and Repovš, D.: Amalgamated products and properly $3$-realizable groups. J. Pure Appl. Algebra 208 (2007), no. 1, 293-296.
  • Cárdenas, M., Lasheras, F.F. and Roy, R.: Direct products and properly $3$-realisable groups. Bull. Austral. Math. Soc. 70 (2004), 199-205.
  • Chiswell, I.M., Collins, D.J. and Huebschmann, J.: Aspherical group presentations. Math. Z. 178 (1981), 1- 36.
  • Cockcroft, W.H.: On two-dimensional aspherical complexes. Proc. London Math. Soc. (3) 4 (1954), 375-384.
  • Dunwoody, M.J.: The accessibility of finitely presented groups. Invent. Math. 81 (1985), 449-457.
  • Dyer, E. and Vasquez, A.T.: Some small aspherical spaces. Collection of articles dedicated to the memory of Hanna Neumann, III. J. Austral. Math. Soc. 16 (1973), 332-352.
  • Funar, L., Lasheras, F.F. and Repovš, D.: Non-compact $3$-manifolds proper homotopy equivalent to geometrically simply connected polyhedra and proper $3$-realizability of groups. Preprint, arXiv:math/0709.1576.
  • Fine, B. and Peluso, A.: Amalgam decomposition for one-relator groups. J. Pure Appl. Algebra 141 (1999), 1-11.
  • Funar, L. and Gadgil, S.: On the geometric simple connectivity of open manifolds. Int. Math. Res. Not. 2004, no. 24, 1193-1248.
  • Funar, L. and Otera, D.E.: Remarks on the WGSC and QSF tameness conditions for finitely presented groups. Preprint, arXiv:math/0610936.
  • Geoghegan, R.: Topological methods in group theory. Graduate Texts in Mathematics 243. Springer, New York, 2008.
  • Geoghegan, R. and Mihalik, M.: Free abelian cohomology of groups and ends of universal covers. J. Pure and Appl. Algebra 36 (1985), 123-137.
  • Geoghegan, R. and Mihalik, M.: The fundamental group at infinity. Topology 35 (1996), no. 3, 655-669.
  • Hog-Angeloni, C., Metzler, W. and Sieradski, A.J., editors: Two-dimensional Homotopy and Combinatorial Group Theory. London Math. Soc. Lecture Notes Series 197. Cambridge Univ. Press, Cambridge, 1993.
  • Karrass, A. and Solitar, D.: Subgroups of HNN groups and groups with one defining relation. Canad. J. Math. 23 (1971), 627-643.
  • Lasheras, F.F.: Universal covers and $3$-manifolds. J. Pure Appl. Algebra 151 (2000), no. 2, 163-172.
  • Lasheras, F.F.: Ascending HNN-extensions and properly $3$-realisable groups. Bull. Austral. Math. Soc. 72 (2005), 187-196.
  • Lyndon, R.C.: Cohomology theory of groups with a single defining relation. Ann. of Math. (2) 52 (1950), 650-665.
  • Lyndon, R.C. and Schupp, P.E.: Combinatorial group theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. Springer-Verlag, Berlin-New York, 1977.
  • Magnus, W.: Über diskontinuierliche Gruppen mit einer definierenden Relation. (Der Freiheitssatz). J. Reine Angew. Math. 163 (1930), 141-165.
  • Magnus, W., Karrass, A. and Solitar, D.: Combinatorial group theory: Presentations of groups in terms of generators and relations. Interscience Publishers, New York-London-Sydney 1966.
  • Mardešić, S. and Segal, J.: Shape Theory. The inverse system approach. North-Holland Mathematical Library 26. North-Holland, Amsterdam-New York, 1982.
  • Mihalik, M.: Semistability at the end of a group extension. Trans. Amer. Math Soc. 277 (1983), no. 1, 307-321.
  • Mihalik, M. and Tschantz, S: One relator groups are semistable at infinity. Topology 31 (1992), no. 4, 801-804.
  • Scott, P. and Wall, C.T.C.: Topological methods in group theory. In Homological group theory (Durham, 1977), 137-203. London Math. Soc. Lecture Note Ser. 36. Cambridge Univ. Press, Cambridge-New York, 1979.