ACTA MATHEMATICA
UNIVERSITATIS COMENIANAE




Vol. LXXX, 1 (2011)
p. 1 - 30

Vertex-transitive maps on a torus

O. Such

Received: November 28, 2008;   Revised: September 30, 2010;   Accepted: November 23, 2010


Abstract.   We examine FVT (free, vertex transitive) actions of wallpaper groups on semiregular tilings. By taking quotients by lattices we then obtain various families of FVT maps on a torus, and describe the presentations of groups acting on the torus. Altogether there are 29 families, 5 arising from the orientation preserving wallpaper groups and 2 from each of the remaining wallpaper groups. We prove that all vertex-transitive maps on torus admit an FVT map structure.

Keywords:  torus, wallpaper group, vertex-transitive map, Cayley map, semiregular tiling 

AMS Subject classification: Primary:  20H15   Secondary: 05C62



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Acta Mathematica Universitatis Comenianae
ISSN 0862-9544   (Printed edition)

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