Hostname: page-component-7479d7b7d-rvbq7 Total loading time: 0 Render date: 2024-07-10T10:41:49.095Z Has data issue: false hasContentIssue false
  • English
  • Français

A Generalization of the Regular Maps of Type{4, 4}b, c and {3, 6}b, c

Published online by Cambridge University Press:  20 November 2018

Dietmar Garbe
Dietmar Garbe*
Affiliation:
Technische Universität Braunschweig, Germany
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In [1], Coxeter gave a complete enumeration of the regular maps on a torus. The maps consist of two families of type {4, 4}b, c and {3, 6}b, c (and their duals). b and c are non-negative integers, which determine the maps uniquely. The maps are irreflexible if and only if bc(b - c) ≠ 0.

On surfaces of genus h > 1, irreflexible regular maps are rather exceptional. The simplest surface of negative characteristic which admits irreflexible regular maps is the orientable surface of genus 7. This was shown by the author [4, Theorem 3. 1 ]. The corresponding map was discovered by J. R. Edmonds [2, p. 388].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 3 , June 1969 , pp. 293 - 297
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Coxeter, H.S.M., Configurations and maps. Reports of a Math. Colloq. (2) 8 (1948) 18–38.Google Scholar
2. Coxeter, H.S.M., Introduction to geometry. (New York, 1961.)Google Scholar
3. Coxeter, H.S.M. and Moser, W. O. J., Generators and relations for discrete groups. (2nd éd., Berlin, 1965.)Google Scholar
4. Garbe, D., Über die regulären Zerlegungen geschlossener orientierbarer Flachen. J. reine angew. Math, (to appear).Google Scholar
5. Sherk, F. A., A family of regular maps of type {6, 6}. Canad. Math. Bull. 5 (1962) 13–20.Google Scholar