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On space form problem

Discrete Geometry Days², Budapest, 9 − 12 July 201 9. On space form problem. Emil Molnár and Jenő Szirmai Budapest University of Technology and Economics , Institute of Mathematics , Department of Geometry Hungary. Abstract On space form problem Emil Molnár–Jenő Szirmai.

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On space form problem

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  1. Discrete Geometry Days², Budapest,9 − 12July 2019 On space form problem Emil Molnár and Jenő Szirmai Budapest University of Technology and Economics, Institute of Mathematics, Department of Geometry Hungary

  2. AbstractOnspaceformproblemEmil Molnár–Jenő Szirmai Discrete Geometry Days², Budapest, 2019

  3. References [1] Molnár, E. (2012). On non-Euclidean crystallography, some football manifolds. StructChem23:1057–1069, DOI:10.1007/s11224-012-0041-z. [2] Molnár, E., Prok, I. and Szirmai, J. (2017). On maximal homogeneous 3-geometries and their realizations. Universe2017, 3, 83; doi: 10.3390/universe3040083, www.mdpi.com/journal/universe [3] Molnár, E. and Szirmai, J. (2017). Top dense hyperbolic ball packings and coverings for complete Coxeterorthoscheme groups. arXiv 1612.04541v1. Publications de l'InstitutMathematique(published by Serbian Academy of Sciences and Arts.Beograd), N.S. 103(117) (2018), 129-146. [4] Molnár, E. and Szirmai, J. (2016). On hyperbolic cobweb manifolds. Studies of the University of Žilina Mathematical Series, 28, 43-52. [5] Molnár, E. and Szirmai, J. (2019). Infinite series of compact hyperbolic manifolds, as possible crystal structures. MatematickiVesnik,toappeararXiv:1711.09799v2 . [6] E. Molnár – J. Szirmai, Hyperbolicspaceformswithcrystallographicapplications and visualization, In: Cocchiarella L. (eds) ICGG 2018 - Proceedings of the 18th International ConferenceonGeometry and Graphics. ICGG 2018.AdvancesinIntelligent Systems and Computing, vol809. Springer, (2018), 320-337, DOI: 10.1007/978-3-319-95588-9_26. [7] Prok, I. (2018). On maximal homogeneous 3-geometries – a polyhedron algorithm for space tilings. Universe2018, 4, 49; doi:10.3390/universe4030049, www.mdpi.com/journal/universe [8] Szirmai, J. (2007). The densest geodesic ball packing by a type of Nil lattices. Beitr. Alg. Geom., (Contr. Alg. Geom.), 48/2, 383–397. Discrete Geometry Days², Budapest, 2019

  4. Repetition: Compact 2-manifoldsThe torus Discrete Geometry Days², Budapest, 2019

  5. Klein-bottle Discrete Geometry Days², Budapest, 2019

  6. Projective plane Discrete Geometry Days², Budapest, 2019

  7. Projectivesphere Discrete Geometry Days², Budapest, 2019

  8. The hyperbolic planeH² and H² x R spacewithvectormodel Discrete Geometry Days², Budapest, 2019

  9. Double torus surface and a hyperbolic tiling Discrete Geometry Days², Budapest, 2019

  10. The 3-torus in E3 Discrete Geometry Days², Budapest, 2019

  11. The truncated octahedron and its Euclidean tilingfor the manifold E3/P212121 Discrete Geometry Days², Budapest, 2019

  12. Complete orthoschemes, t=1. Discrete Geometry Days², Budapest, 2019

  13. Complete orthoschemes, t=1. Discrete Geometry Days², Budapest, 2019

  14. Coxeter-Schläfli matrix and its inverse Discrete Geometry Days², Budapest, 2019

  15. The volumes of complete orthoschemes, by the ideas of N. I. Lobachevsky Discrete Geometry Days², Budapest, 2019

  16. E. Molnár, Two hyperbolic football manifolds. In: Proceedings of International Conference on Differential Geometry and Its Applications, Dubrovnik Yugoslavia, 1988. 217–241. E. Molnár, On non-Euclidean crystallography, some football manifolds, Struct Chem (2012) 23:1057–1069 DOI 10.1007/s11224-012-0041-z Discrete Geometry Days², Budapest, 2019

  17. Ball packings and coverings,optimal densities with Case 1.s.i.a Discrete Geometry Days², Budapest, 2019

  18. Discrete Geometry Days², Budapest, 2019

  19. To the football manifold {5, 6, 6}generators and relations, H1 = Z14 Discrete Geometry Days², Budapest, 2019

  20. Twofold covering for M, a dodecahedron manifold M1 Discrete Geometry Days², Budapest, 2019

  21. Construction of cobweb manifolds on the base of 2-truncated symmetric orthoschemes, u = v = w (= 6 = 2z, z odd ) Discrete Geometry Days², Budapest, 2019

  22. Construction of cobweb (tube) manifolds Discrete Geometry Days², Budapest, 2019

  23. Construction of cobweb (tube) manifolds Discrete Geometry Days², Budapest, 2019

  24. Cobweb (tube) manifold Cw(6, 6, 6) Discrete Geometry Days², Budapest, 2019

  25. Construction of cobweb (tube) manifolds Discrete Geometry Days², Budapest, 2019

  26. Discrete Geometry Days², Budapest, 2019

  27. Construction of cobweb (tube) manifolds with new fundamental domain Discrete Geometry Days², Budapest, 2019

  28. Discrete Geometry Days², Budapest, 2019

  29. Construction of cobweb (tube) manifolds with new fundamental domain Discrete Geometry Days², Budapest, 2019

  30. Construction of cobweb (tube) manifolds with concave fundamental domain Discrete Geometry Days², Budapest, 2019

  31. Construction of cobweb (tube) manifolds with concave fundamental domain Discrete Geometry Days², Budapest, 2019

  32. Discrete Geometry Days², Budapest, 2019

  33. Discrete Geometry Days², Budapest, 2019

  34. Discrete Geometry Days², Budapest, 2019

  35. L Discrete Geometry Days², Budapest, 2019

  36. Space forms on {4, 6, 8} Theorem.The Archimedean solid {4, 6, 8} can be endowed with 2 hyperbolic space form structures at least G and H, each with two face pairing generators at the starting directed edges . So we get two fixed-point-free fundamental groups: G := (s1, s | 1 = s13s-2s1ss1ss1s-2 = s3s1s-2s1s2s11s-1s1-1s2s1s2s1); withAbelianization 1 = s16s-2 = s12s2, i.e.H1(G) = G/[G, G] = Z8 × Z6 ; is the first homology group; andH := (s, s1 | 1 = s1ss1-1ss1-1s2s1-1ss1-1ss1s2 = s1ss1-1s-1s1s-1s1s-1s1-1ss1s3), with Abelianization 1 = s8s1-2 = s2s12, i.e.H1(H) = H/[H, H] = Z10 × Z2, respectively. The ball packing and covering densities are ≈ 0,57313 resp. ≈ 1,44528 (by J. Szirmai). Discrete Geometry Days², Budapest, 2019

  37. Discrete Geometry Days², Budapest, 2019

  38. Discrete Geometry Days², Budapest, 2019

  39. The {4, 6, 10}hyperbolic space form (above) with 3 generators and 3 relations Theorem. The above (with starting edge triple 1 and 7, and generators a, b, c ) {4, 6, 10} manifold construction closes. Its fundamental group Г has a presentationГ = (a, b, c | 1 = a-1b-1c-1cbab-2c-2abacba-1c-1b-2c-1 = abac-1abacb2cabcabc2b2a-1c-1b-2c-2abacb2cabc =ab-1ab-2c-2abacab-2c-2ac-1abacb2cabcabcabacb2c).Its Abelianization leads to1 = b-2c-2 =a8b10c4 = a12b4c2,i.e. to the first homology groupH1(Г) = Z28 × Z6 × Z2.The ball packing and covering densities are ≈ 0,55128 and ≈ 1,43868, respectively (by the computation of Jenő Szirmai). Discrete Geometry Days², Budapest, 2019

  40. OtherThurstongeometries,Niltranslations 2020.01.05. Discrete Geometry Days², Budapest, 2019

  41. Niltranslations and invariantRiemannmetric Discrete Geometry Days², Budapest, 2019

  42. The densest (?!, 14-neighbours) ball packing in Nil geometry Picture by Benedek Schultz with top density 0.780845 by Jenő Szirmai Discrete Geometry Days², Budapest, 2019

  43. Thank you Discrete Geometry Days², Budapest, 2019

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