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Tiling the torus and other space forms


We consider graphs on two-dimensional space forms which are quotient graphs Γ/F, where Γ is an infinite, 3-connected, face, vertex, or edge transitive planar graph andF is a subgroup of Aut(Γ), all of whose elements act freely on Γ. The enumeration of quotient graphs with transitivity properties reduces to computing the normalizers in Aut(Γ) of the subgroupsF. Results include: all isogonal toriodal polyhedra belong to the two families found by Grünbaum and Shephard; there are no transitive graphs on the Möbius band; there is a graph on the Klein bottle whose automorphism group acts transitively on its faces, edges, and vertices.


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This paper is an expanded version of a lecture presented to the Conference on Combinatorial Geometry, Oberwolfach, Germany, September 1984.

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Senechal, M. Tiling the torus and other space forms. Discrete Comput Geom 3, 55–72 (1988).

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  • Space Form
  • Klein Bottle
  • Invariant Subgroup
  • Transitivity Property
  • Quotient Graph