Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T14:19:18.944Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Asphericity of a Family of Positive Relative Group Presentations

Part of: Special aspects of infinite or finite groups Low-dimensional topology

Published online by Cambridge University Press:  05 January 2017

Suzana Aldwaik  and
Martin Edjvet
Suzana Aldwaik
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK ([email protected])
Martin Edjvet
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Excluding four exceptional cases, we determine the asphericity of the relative presentation for m ⩾ 2. If H = ⟨g, h⟩ ⩽ G, then the exceptional cases occur when H is isomorphic to C5 or C6.

Keywords

relative group presentationspicturesasphericity

MSC classification

Secondary: 20F05: Generators, relations, and presentations 57M05: Fundamental group, presentations, free differential calculus
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 3 , August 2017 , pp. 545 - 564
Copyright
Copyright © Edinburgh Mathematical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Ahmad, A., Al-Mulla, M. and Edjvet, M., Asphericity of length four relative group presentations, J. Alg. Applic., in press.Google Scholar
2. Baik, Y. G., Bogley, W. A. and Pride, S. J., On the asphericity of length four relative group presentations, Int. J. Alg. Comput. 7 (1997), 227–312.CrossRefGoogle Scholar
3. Bogley, W. A. and Pride, S. J., Aspherical relative presentations, Proc. Edinb. Math. Soc. 35 (1992), 1–39.CrossRefGoogle Scholar
4. Bogley, W. A. and Williams, G., Efficient finite groups arising in the study of relative asphericity, Math. Z. 284 (2016), 507–535.CrossRefGoogle Scholar
5. Collins, D. J. and Huebschmann, J., Spherical diagrams and identities among relations, Math. Annalen 261 (1982), 155–183.CrossRefGoogle Scholar
6. Davidson, P. J., On the asphericity of a family of relative group presentations, Int. J. Alg. Comput. 19 (2009), 159–189.CrossRefGoogle Scholar
7. Edjvet, M., On the asphericity of one-relator relative presentations, Proc. R. Soc. Edinb. A124 (1994), 713–728.CrossRefGoogle Scholar
8. Howie, J. and Metaftsis, V., On the asphericity of length five relative group presentations, Proc. Lond. Math. Soc. 82 (2001), 173–194.CrossRefGoogle Scholar
9. Levin, F., Solutions of equations over groups, Bull. Am. Math. Soc. 68 (1962), 603–604.CrossRefGoogle Scholar
10. Metaftsis, V., On the asphericity of relative group presentations of arbitrary length, Int. J. Alg. Comput. 13 (2003), 323–339.CrossRefGoogle Scholar