DOI QR코드

DOI QR Code

The Geometry Descriptions of Crystallographic Groups of Sol41

  • Yoo, Won Sok (Department of Applied Mathematics, Kumoh National Institute of Technology)
  • 투고 : 2017.04.09
  • 심사 : 2017.06.25
  • 발행 : 2017.06.30

초록

The connected and simply connected four-dimensional matrix solvable Lie group $Sol^4_1$ is the four-dimensional geometry. A crystallographic group of $Sol^4_1$ is a discrete cocompact subgroup of $Sol^4_1{\rtimes}D(4)$. In this paper, we geometrically describe the crystallographic groups of $Sol^4_1$.

키워드

참고문헌

  1. R. Filipkiewicz, "Four dimensional geometries", PhD diss, University of Warick, 1983.
  2. C. T. C. Wall, "Geometric structures on compact complex analytic surfaces", Topology, Vol. 25, pp. 119-153, 1986. https://doi.org/10.1016/0040-9383(86)90035-2
  3. J. H. Jo and J. B. Lee, "Nielsen type numbers and homotopy minimal periods for maps on solvmanifolds with $Sol_1^4$-geometry", Fixed Point Theory and Applications, Vol. 175, pp. 1-15, 2015.
  4. J. B. Lee and X. Zhao, "Nielsen type numbers and homotopy minimal periods for all continuous maps on the 3-nilmanifolds", Sci. China Ser. A Math., Vol. 51, pp. 351-360, 2008. https://doi.org/10.1007/s11425-008-0003-5
  5. J. B. Lee and X. Zhao, "Nielsen type numbers and homotopy minimal periods for maps on the 3-solvmanifolds", Algebraic & Geometric Topology, Vol. 8, pp. 563-580, 2008. https://doi.org/10.2140/agt.2008.8.563
  6. K. Y. Ha and J. B. Lee, "Crystallographic groups of Sol", Mathematische Nachrichten, Vol. 286, pp. 1614-1667, 2013. https://doi.org/10.1002/mana.201200304
  7. K. B. Lee and S. Thuong, "Infra-solvmanifolds of $Sol_1^4$", J. Korean Math. Soc., Vol. 52, pp. 1209-1251, 2015. https://doi.org/10.4134/JKMS.2015.52.6.1209
  8. K. Y. Ha and J. B. Lee, "The $R_{\infty}$ property for crystallographic groups of Sol", Topol. Appl., Vol. 181, pp. 112-133, 2015. https://doi.org/10.1016/j.topol.2014.12.005
  9. K. Y. Ha, J. B. Lee, and P. penninckx, "Formulas for the reidemeister, lefschetz and nielsen coincidence number of maps between infra-nilmanifolds", Fixed Point Theory Applications, Vol. 39, pp. 1-23, 2012.
  10. P. R. Health, "Groupoids and relations among reidemeister and among nielsen numbers", Topol. Appl., Vol. 181, pp. 3-33, 2015. https://doi.org/10.1016/j.topol.2014.08.033