F-actions and parallel-product decomposition of reflexible maps
Alen Orbanić
University of Ljubljana Faculty of Mathematics and Physics, Department of Mathematics Jadranska 21 1000 Ljubljana Slovenia
DOI: 10.1007/s10801-007-0069-0
Abstract
The parallel product of two rooted maps was introduced by S.E. Wilson in 1994. The main question of this paper is whether for a given reflexible map M one can decompose the map into a parallel product of two reflexible maps. This can be achieved if and only if the monodromy (or the automorphism) group of the map has at least two minimal normal subgroups. All reflexible maps up to 100 edges, which are not parallel-product decomposable, are calculated and presented. For this purpose, all degenerate and slightly-degenerate reflexible maps are classified.
In this paper the theory of F-actions is developed including a classification of quotients and parallel-product decomposition. Projections and lifts of automorphisms for quotients and for parallel products are studied. The theory can be immediately applied on rooted maps and rooted hypermaps as they are special cases of F-actions.
Pages: 507–527
Keywords: keywords rooted map; $F$-action; map quotients; normal quotient; parallel product; reflexible map; parallel-product decomposition
Full Text: PDF
References
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2. Bergau, P., & Garbe, D. (1989). Non-orientable and orientable regular maps. In Proceedings of groups-Korea 1998, Lecture notes in mathematics (Vol. 1398, pp. 29-42). New York: Springer.
3. Bosma, W., Cannon, C., & Playoust, C. (1997). The MAGMA algebra system I: the user language. Journal of Symbolic Computation, 24, 235-265.
4. Brahana, H. R. (1927). Regular maps and their groups. American Journal of Mathematics, 49, 268- 284.
5. Breda d'Azevedo, A., & Nedela, R. (2001). Join and intersection of hypermaps. Acta Universitatis M. Belii Series Mathematics, 9, 13-28.
6. Conder, M. http://www.math.auckland.ac.nz/~conder/.
7. Conder, M., & Dobcsányi, P. (2001). Determination of all regular maps of small genus. Journal of Combinatorial Theory Series B, 81, 224-242.
8. Conder, M., & Dobcsányi, P. (1999). Computer program LOWX, censuses of rotary maps. http://www. math.auckland.ac.nz/~peter.
9. Coxeter, H. S. M., & Moser, W. O. J. (1984). Generators and relations for discrete groups (4th ed.). Berlin: Springer.
10. Ferri, M. (1976). Una rappresentazione delle n-varieta topologiche triangolabili mediante grafi (n + 1)-colorati. Bollettino Unione Matematica Italiana Sezione B (5), 13(1), 250-260.
11. Garbe, D. (1969). Über die regulären Zerlegungen geschlossener orientierbarer Flächen. Journal für die Reine und Angewandte Mathematik, 237, 39-55.
12. Hartley, M. I. (1999). All polytopes are quotients, and isomorphic polytopes are quotients by conjugate subgroups. Discrete & Computation Geometry, 21(2), 289-298.
13. Hartley, M. I. (1999). More on quotient polytopes. Aequationes Mathematicae, 57, 108-120.
14. Hodoš\check cek, M., Borštnik, U., & Janeži\check c, D. (2002). Crow for large scale macromolecular simulations. Cell. Mol. Biol. Letters, 7(1), 118-119.
15. Jones, G. A., & Thornton, J. S. (1983). Operations on maps, and outer automorphisms. Journal of Combinatorial Theory Series B, 35(2), 93-103.
16. Li, C. H., & Širá\check n, J. (2005). Regular maps whose groups do not act faithfully on vertices, edges, or faces. European Journal of Combinatorics, 26(3-4), 521-541.
17. Lins, S. (1982). Graph-encoded maps. Journal of Combinatorial Theory Series B, 32(2), 171-181.
18. Malni\check c, A., Nedela, R., & Škoviera, M. (2002). Regular homomorphisms and regular maps. European Journal of Combinatorics, 23, 449-461.
19. Orbanić, A. (2005). Parallel-product decomposition of edge-transitive maps. http://arxiv.org/PS_ cache/math/pdf/0510/0510428.pdf.
20. Orbanić, A. (2006). Edge-transitive maps. Doctoral dissertation, University of Ljubljana, Ljubljana.
21. Richter, R. B., Širá\check n, J., Jajcay, R., Tucker, T. W., & Watkins, M. E. (2005). Cayley maps. Journal of Combinatorial Theory Series B, 95, 189-245.
22. Sherk, F. A. (1959). The regular maps on a surface of genus three. Canadian Journal of Mathematics, 11, 452-480.
23. Širá\check n, J., Tucker, T. W., & Watkins, M. E. (2001). Realizing finite edge-transitive orientable maps. Journal of Graph Theory, 37, 1-34.
24. Tucker, T. W.: Private communication.
25. Vince, A. (1983). Combinatorial maps. Journal of Combinatorial Theory Series B, 34(1), 1-21.
26. Wilson, S. E. (1976). New techniques for the construction of regular maps. Doctoral dissertation, University of Washington, Seattle.
27. Wilson, S. E. (1978). Riemann surfaces over regular maps. Canadian Journal of Mathematics, 30(4), 763-782.
28. Wilson, S. E. (1985). Bicontactual regular maps. Pacific Journal of Mathematics, 120(2), 437-451.
29. Wilson, S. E. (1994). Parallel products in groups and maps. Journal of Algebra, 167, 539-546.
30. Wilson, S. E. (2002). Families of regular graphs in regular maps. Journal of Combinatorial Theory Series B, 85(2), 269-289.
31. Wilson, S. E. (2005). Wilson's census of rotary maps. http://www.ijp.si/RegularMaps/.
32. Zvonkin, A. (2001). Megamaps: construction and examples. In Discrete mathematics and theoretical computer science proceedings (pp. 329-340). Paris: MIMD.
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