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Regular incidence quasi-polytopes and regular maps

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Abstract

We define incidence quasi-polytopes and give a procedure for constructing regular incidence quasi-polytopes. We use this procedure to construct a finite map of type {a,b} for all even a and b, and infinitely many such maps when a or b is divisible by 4 and both are greater than or equal to 4.

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References

  1. Brahana, H. R., ‘Regular Maps on an Anchor Ring’,Amer. J. Math. 48 (1926), 225–240.

    Google Scholar 

  2. Brahana, H. R., ‘Regular Maps and their Groups’,Amer. J. Math. 49 (1927), 268–284.

    Google Scholar 

  3. Buekenhout, F., ‘Diagrams for Geometries and Groups’,J. Comb. Theory A27 (1979), 121–151.

    Google Scholar 

  4. Coxeter, H. S. M.,Regular Polytopes (3rd edn), Dover, New York, 1973.

    Google Scholar 

  5. Coxeter, H. S. M. and Moser, W. O. J.,Generators and Relations for Discrete Groups (4th edn), Springer, Berlin, 1980.

    Google Scholar 

  6. Danzer, L., ‘Regular Incidence-Complexes and Dimensionally Unbounded Sequence of Such, I’,Proc. Int. Conf. Convexity and Graph Theory, Israel, 1981.

  7. Danzer, L. and Schulte, E., ‘Reguläre Inzidenzkomplexe I’,Geom. Dedicata 13-(1982), 295–308.

    Google Scholar 

  8. Dress, A., ‘Presentations of Discrete Groups, Acting on Simply Connected Manifolds, in Terms of Parametrized Systems of Coxeter Matrices — a Systematic Approach’,Advances in Mathemetics 63 (1987), 196–212.

    Google Scholar 

  9. Grünbaum, B., ‘Regular Polyhedra — Old and New’,Aequationes Math. 16 (1977), 1–20.

    Google Scholar 

  10. Grünbaum, B., ‘Regularity of Graphs, Complexes and Designs’,Coll. Int. C.N.R.S. No. 260, Problemes Combinatorie et Théorie des Graphes, Orsay, 1977, pp. 191–197.

  11. McMullen, P., ‘Combinatorially Regular Polytopes’,Mathematika 14 (1967), 142–150.

    Google Scholar 

  12. Schulte, E., ‘Reguläre Inzidenzkomplexe II’,Geom. Dedicata 14 (1983), 33–56.

    Google Scholar 

  13. Schulte, E., ‘Reguläre Inzidenzkomplexe III’,Geom. Dedicata 14 (1983), 57–79.

    Google Scholar 

  14. Tits, J., ‘A Local Approach to Buildings’,The Geometric Vein (C. Davis, B. Grünbaum, and F. A. Sherk (eds)), Springer, New York, 1982, 519–547.

    Google Scholar 

  15. Vince, A., ‘Combinatorial Maps’,J. Comb. Theory B34 (1983), 1–21.

    Google Scholar 

  16. Vince, A., ‘Regular Combinatorial Maps’,J. Comb. Theory B35 (1983), 256–277.

    Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Sociology, 326 Lincoln Hall, University of Illinois at Urbana-Champaign, 702 S. Wright St., 61801, Urbana, IL, U.S.A.

    Adam Stephanides

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  1. Adam Stephanides
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Stephanides, A. Regular incidence quasi-polytopes and regular maps. Geom Dedicata 30, 211–221 (1989). https://doi.org/10.1007/BF00181553

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  • DOI: https://doi.org/10.1007/BF00181553

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