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The Regular Maps on a Surface of Genus Three

  • F. A. Sherk (a1)

Extract

A considerable volume of research on the theory of regular maps is now in existence. Systematic enumerations of regular maps on the surfaces of genus 1 and 2 were begun by Brahana (1; 2) and completed by Coxeter (6; 7, p. 141). In addition Coxeter enumerated the regular maps on the simplest non-orientable surfaces (7, pp. 116, 139), and constructed tables of some interesting families of regular maps (3; 7, p. 140).

Most of the regular maps on a surface of genus 3 have appeared in these papers, but no systematic enumeration of them seems to have been attempted. The ultimate goal of this paper is a complete list of these regular maps. However, the families of maps {j.p,q}and {j.p,j.q} which are defined in § 4 and listed in Tables I and II are of considerable interest in themselves. Also of some importance is the complete list of regular maps of type {p,3} with six or fewer faces (§ 5 and Table III).

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References

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1. Brahana, H.R., Regular maps on an anchor ring, Amer. J. Math., 48 (1926), 225-40.
2. Brahana, H.R., Regular maps and their groups, Amer. J. Math., 49 (1927), 268-84.
3. Coxeter, H. S. M., Regular skew polyhedra in three and four dimensions and their topological analogues, Proc. London Math. Soc. (2), 43 (1937), 3362.
4. Coxeter, H. S.M., The abstract groups Gm'n'?, Trans. Amer. Math. Soc, 45 (1939), 73150.
5. Coxeter, H. S.M., Regular polytopes (London, 1948).
6. Coxeter, H. S.M., Configurations and maps, Reports of a Math. Colloq. (2), 8 (1948), 1838.
7. Coxeter, H. S.M., and Moser, W. O. J., Generators and relations for discrete groups, Ergebn. Math., 14 (1957).
8. Dyck, W., Notiz ueber eine regulàre Riemann'sche Flache vom geschlechte drei und die zugehörige “normalcurve” vierter ordnung, Mat. Ann., 17 (1880), 510–16.
9. Frucht, R., A one-regular graph of degree three, Can. J. Math., 4 (1952), 240–7.
10. Klein, F., Ueber die transformationen siebenter ordnung der elliptischen functionen, Mat. Ann., 14 (1879), 428-71.
11. Klein, F., Lectures on the icosahedron and the solution of equations of the fifth deegree trans. G. C. Morrice (London, 1913).
12. Shephard, G. C., Regular complex polytopes, Proc. London Math. Soc. (3), 2 (1952), 8297.
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The Regular Maps on a Surface of Genus Three

  • F. A. Sherk (a1)

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