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ON THE CRYSTALLOGRAPHIC GROUP OF Sol4m,n

  • Yoo, Won Sok (Department of Applied Mathematics Kumoh National Institute of Technology)
  • Received : 2020.12.07
  • Accepted : 2020.12.28
  • Published : 2021.02.15

Abstract

The purpose of this paper is to determine the structure of the crystallographic group 𝚷 of the 4-dimensional solvable Lie group Sol 4m,n that the translation subgroup of 𝚷, 𝚪 := 𝚷; ∩ Sol4m,n, is generated by the particular elements.

Keywords

Acknowledgement

This research was supported by Kumoh National Institute of Technology (202001960001).

References

  1. K. Dekimpe, K. B. Lee and F. Raymond, Bieberbach theorems for solvable Lie groups, Asian J. Math., 5 (2001), 499-508. https://doi.org/10.4310/AJM.2001.v5.n3.a6
  2. K. Y. Ha and J. B. Lee, Crystallographic groups of Sol, Math. Nachr., 286 (2013), 1614-1667. https://doi.org/10.1002/mana.201200304
  3. J. A. Hillman, Four-manifolds , Geometries and Knots, Geometry & Topology Monographs, 5, Geometry & Topology Publications, Coventry, 2002.
  4. K. B. Lee and S. Thuong, Infra-solvmanifolds of Sol41, J. Korean Math. Soc., 52 (2015), 1209-1251. https://doi.org/10.4134/JKMS.2015.52.6.1209
  5. J. B. Lee, K. B. Lee, J. Shin and S. Yi, Unimodular groups of type ${\mathbb{R}}^3{\times}{\mathbb{R}}$, J. Korean Math. Soc., 44 (2007), 1121-1137. https://doi.org/10.4134/JKMS.2007.44.5.1121
  6. P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc., 15 (1983), 401-487. https://doi.org/10.1112/blms/15.5.401
  7. W. P. Thurston, Three-dimensional Geometry and Topology, 1, Princeton University Press, Princeton, NJ, 1997.
  8. S. V. Thuong, Classification of closed manifolds with Sol41-geometry, Geom. Dedicata., 199 (2019), 373-397. https://doi.org/10.1007/s10711-018-0354-1
  9. C. T. C. Wall, Geometric structures on compact complex analytic surfaces, Topology, 25 (1986), 119-153. https://doi.org/10.1016/0040-9383(86)90035-2