On infinite regular and chiral maps

Preprint English OPEN
Arredondo, John A.; Valdez, Camilo Ramírez y Ferrán;
(2015)
  • Subject: Mathematics - Combinatorics | 52C20, 52B15
    arxiv: High Energy Physics::Lattice | Computer Science::Databases

We prove that infinite regular and chiral maps take place on surfaces with at most one end. Moreover, we prove that an infinite regular or chiral map on an orientable surface with genus can only be realized on the Loch Ness monster, that is, the topological surface of i... View more
  • References (16)
    16 references, page 1 of 2

    [CPR+15] Thierry Coulbois, Daniel Pellicer, Miguel Raggi, Camilo Ramírez, and Ferrán Valdez, The topology of the minimal regular covers of the Archimedean tessellations, Adv. Geom. 15 (2015), no. 1, 77-91.

    [Cox] Harold S. M. Coxeter, Regular Skew Polyhedra in Three and Four Dimension, and their Topological Analogues, Proc. London Math. Soc. S2-43, no. 1, 33.

    [Fre31] Hans Freudenthal, Über die Enden topologischer Räume und Gruppen, Math. Z. 33 (1931), no. 1, 692-713 (German).

    [Ghy95] Étienne Ghys, Topologies des Feullies Generiques, Ann. of Math. (2) 141 (1995), no. 2, 387-422 (French).

    [JS78] Gareth A. Jones and David Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978), no. 2, 273-307.

    [Ker23] Bela Kerékjártó, Vorlesungen über Topologie. I, Springer, Berlin, 1923.

    [Pel12] Daniel Pellicer, Developments and open problems on chiral polytopes, Ars Math. Contemp. 5 (2012), no. 2, 333-354.

    [Ray60] Frank Raymond, The end point compactification of manifolds, Pacific J. Math. 10 (1960), 947-963.

    [Ser03] Jean-Pierre Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation.

    [Sta68] John R. Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. (2) 88 (1968), 312-334.

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