Abstract
This is a survey on the tilings (T, Γ) in the title where the vertex stabilizers in are finite spherical S2 or infinite Euclidean E2 (cocompact) plane groups. The results are collected in figures and tables and illustrated by an infinite family series Family 30 in Section 4. The obtained orbifolds, maybe after splitting procedure, are realized in seven homogeneous Riemannian 3-spaces by means of projective metrics.
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Bibliography
Coxeter, H. S. M. — Moser, W. O. J. (1980) Generators and relations for discrete groups. 4th ed. Springer, Berlin-Heidelberg-New York.
Dress, A. W. M. (1987) Presentation of discrete groups acting on simply connected manifolds in terms of parametrized systems of Coxeter matrices. Advances in Math.63 196–212.
Dress, A. W. M. — Huson, D. H. — Molnár, E. (1993) The classification of face-transitive periodic three-dimensional tilings. Acta Crystallographica, A 49 806–817.
Huson, D. H. (1993) The generation and classification of tile-κ-transitive tilings of the Euclidean plane, the sphere and the hyperbolic plane. Geometriae Dedicata, 47 269–296.
Lučić, Z. — MOLNÁR, E. (1990) Combinatorial classification of fundamental domains of finite area for planar discontinuous isometry groups. Arch. Math.54 511–520.
Molnár, E. (1992) Polyhedron complexes with simply transitive group actions and their realizations. Acta Math. Hungarica, 59 175–216.
Molnár, E. (1996) Discontinuous groups in homogeneous Riemannian spaces by classification of D-symbols. Publ. Math. Debrecen, 49/3–4 265–294.
Molnár, E. (1997) The projective interpretation of the eight 3-dimensional homogeneous geometries. Beiträge zur Algebra und Geometric (Contributions to Algebra and Geometry), 38 No. 2, 261–288.
Molnár, E. — Prok, I. (1988) A polyhedron algorithm for finding space groups. Proc. of Third Int. Conf. on Engineering Graphics and Descriptive Geometry, Vienna Vol.2, pp. 37–44.
Molnár, E. — Prok, I. (1994) Classification of solid transitive simplex tilings in simply connected 3-spaces, Part I. The combinatorial description by figures and tables, results in spaces of constant curvature. Colloquia Math. Soc. János Bolyai63. Intuitive Geometry, Szeged (Hungary), 1991. North-Holland Publ. Comp. Amsterdam-Oxford-New York, 311–362.
Molnár, E. — Prok, I. — Szirmai, J. (1997) Classification of solid transitive simplex tilings in simply connected 3-spaces, Part II. Metric realizations of the maximal simplex tilings. Periodica Math. Hungar.35/1–2 47–94.
Molnár, E. — Prok, I. — Szirmai, J. (1998) Two families of fundamental tilings and their realizations in various 3-spaces. Proc. of International Scientific Conferences of Mathematics, Žilina, Slovakia, Vol.2 43–64.
Molnár, E. — Szirmai, J. — Weeks, J. R. 3-simplex tilings, splitting orbifolds and manifolds. Annales Univ. Sci. Budapest, Sect. Math. (submitted)
Molnár, E. — Szirmai, J. Minimally presented orientable splitting 3-manifold with one cusp. Annales Univ. Sci. Budapest, Sect. Math. (submitted)
Prok, I. (1999) Fundamental tilings in spaces of constant curvature with regular polyhedra (in Hungarian). PhD thesis, Budapest University of Technology and Economics, Faculty of Natural Sciences.
Scott, P. (1983) The geometry of 3-manifolds. Bull. London Math. Soc., 15 401–487. Russian translation: Moscow ‘Mir’ (1986).
Stojanović, M. (1993) Some series of hyperbolic space groups. Annales Univ. Sci. Budapest, Sect. Math.36 85–102.
Szirmai, J. (1996) Metrische Realisierungen von zwei Familien der dreidimensionalen körpertransitiven Simplexpflasterungen, Annales Univ. Sci. Budapest, Sect. Math.39 145–162.
Thurston, W. P. (1997) Three-dimensional geometry and topology I. Ed. by SILVIO LEVY, Princeton Univ. Press, Princeton, New Jersey.
Zhuk, I. K. (1983) Fundamental tetrahedra in Euclidean and Lobachevsky spaces. Soviet Math. Dokl.27 540–543.
Zieschang, H.— Vogt, E. — Coldewey, H. D. (1980) Surfaces and planar discontinuous groups. Lecture Notes in Math. 835, Springer, Berlin-Heidelberg-New York.
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Molnár, E., Prok, I., Szirmai, J. (2006). Classification of Tile-Transitive 3-Simplex Tilings and Their Realizations in Homogeneous Spaces. In: Prékopa, A., Molnár, E. (eds) Non-Euclidean Geometries. Mathematics and Its Applications, vol 581. Springer, Boston, MA. https://doi.org/10.1007/0-387-29555-0_17
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