East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
권호 표지
DOI QR코드

DOI QR Code

REMARKS ON GROUP EQUATIONS AND ZERO DIVISORS OF TOPOLOGICAL STRUCTURES

  • Seong-Kun Kim (Division of Liberal Studies, Kangwon National University)
  • Received : 2023.05.08
  • Accepted : 2023.05.25
  • Published : 2023.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

The motivation in this paper comes from the recent results about Bell inequalities and topological insulators from group theory. Symmetries which are interested in group theory could be mainly used to find material structures. In this point of views, we study group extending by adding one relator which is easily called an equation. So a relative group extension by a adding relator is aspherical if the natural injection is one-to-one and the group ring has no zero divisor. One of concepts of asphericity means that a new group by a adding relator is well extended. Also, we consider that several equations and relative presentations over torsion-free groups are related to zero divisors.

Keywords

References

  1. W.A. Bogley, M. Edjvet and G. Williams, Aspherical relative presentations all over again, Groups St Andrews in Birmingham, London Math. Soc.Lecture Note Series, (2019) 169-199.
  2. W.A. Bogley and S.J. Pride, Aspherical relative presentations, Proc. Edinburgh Math. Soc. 35,(1992), 1-39. https://doi.org/10.1017/S0013091500005290
  3. S. D. Brodskii and James Howie One-relator products of torsion-free groups, Glasgow Math. J. 35, (1993), 99-104. https://doi.org/10.1017/S0017089500009617
  4. A. Clifford and R. Z. Goldstein, Tesselations of S2 and equations over torsion-free groups, Proc. Edinb. Math. Soc. 38, (1995), 485-493. https://doi.org/10.1017/S0013091500019283
  5. M. Edjvet, Equations over groups and a theorem of Higman, Neumann, and Neumann, Proc. Lond. Math. Soc. 62, (1991), 563-589. https://doi.org/10.1112/plms/s3-62.3.563
  6. J. Howie, On pairs of 2-complexes and systems of equations groups, J. Reine Angew. Math. 324, (1981), 165-174. https://doi.org/10.1515/crll.1981.324.165
  7. S. V. Ivanov and A. A. Klyachko, Solving equations of length at most six over torsion-free groups, J. Group Theory 3, (2000), 329-337. https://doi.org/10.1515/jgth.2000.026
  8. Seong K. Kim, On the asphericity of certain relative presentations over torsion-free groups, Int. J. Algebra Comput., 18(6) (2008), 979-987. https://doi.org/10.1142/S0218196708004718
  9. V. Uur Gney and Mark Hillery, Phys. Rev. A 90, 062121 (2016).