Cellular automorphisms and self-duality

Abrams, Lowell; Slilaty, Daniel

doi:10.1090/tran/6258

Cellular automorphisms and self-duality
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by Lowell Abrams and Daniel Slilaty PDF

Trans. Amer. Math. Soc. 367 (2015), 7695-7773 Request permission

Abstract:

We catalog up to a type of reducibility all cellular automorphisms of the sphere, projective plane, torus, Klein bottle, and three-crosscaps (Dyck’s) surface. We also show how one can obtain all self-dual embeddings in a surface $S$ given a catalog of all irreducible cellular automorphisms in $S$.

References

Amos Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201–217. MR 321797, DOI 10.1016/S0012-365X(73)80002-0
Mark S. Anderson and R. Bruce Richter, Self-dual Cayley maps, European J. Combin. 21 (2000), no. 4, 419–430. MR 1756148, DOI 10.1006/eujc.1999.0357
Dan Archdeacon, A survey of self-dual polyhedra, Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity (Prachatice, 1990) Ann. Discrete Math., vol. 51, North-Holland, Amsterdam, 1992, pp. 5–12. MR 1206236, DOI 10.1016/S0167-5060(08)70598-3
Dan Archdeacon, Self-dual embeddings of complete multipartite graphs, J. Graph Theory 18 (1994), no. 7, 735–749. MR 1297194, DOI 10.1002/jgt.3190180709
Dan Archdeacon, Pavol Gvozdjak, and Jozef Širáň, Constructing and forbidding automorphisms in lifted maps, Math. Slovaca 47 (1997), no. 2, 113–129. MR 1476862
Dan Archdeacon and Nora Hartsfield, Self-dual embeddings of complete bipartite graphs, J. Combin. Theory Ser. B 54 (1992), no. 2, 249–256. MR 1152452, DOI 10.1016/0095-8956(92)90056-4
Dan Archdeacon and Seiya Negami, The construction of self-dual projective polyhedra, J. Combin. Theory Ser. B 59 (1993), no. 1, 122–131. MR 1234388, DOI 10.1006/jctb.1993.1059
Dan Archdeacon and R. Bruce Richter, The construction and classification of self-dual spherical polyhedra, J. Combin. Theory Ser. B 54 (1992), no. 1, 37–63. MR 1142263, DOI 10.1016/0095-8956(92)90065-6
Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281
Béla Bollobás and Oliver Riordan, Percolation, Cambridge University Press, New York, 2006. MR 2283880, DOI 10.1017/CBO9781139167383
Thomas Breuer, Characters and automorphism groups of compact Riemann surfaces, London Mathematical Society Lecture Note Series, vol. 280, Cambridge University Press, Cambridge, 2000. MR 1796706
S. Allen Broughton, Classifying finite group actions on surfaces of low genus, J. Pure Appl. Algebra 69 (1991), no. 3, 233–270. MR 1090743, DOI 10.1016/0022-4049(91)90021-S
John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss, The symmetries of things, A K Peters, Ltd., Wellesley, MA, 2008. MR 2410150
J. H. Conway and D. H. Huson, The orbifold notation for two-dimensional groups, Structural Chemistry 13 (2002), no. 3/4, 247–257.
H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 4th ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 14, Springer-Verlag, Berlin-New York, 1980. MR 562913, DOI 10.1007/978-3-662-21943-0
M. Dehn, Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938), no. 1, 135–206 (German). Das arithmetische Feld auf Flächen. MR 1555438, DOI 10.1007/BF02547712
Peter Giblin, Graphs, surfaces and homology, 3rd ed., Cambridge University Press, Cambridge, 2010. MR 2722281, DOI 10.1017/CBO9780511779534
Jonathan L. Gross and Thomas W. Tucker, Topological graph theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1987. A Wiley-Interscience Publication. MR 898434
Akikazu Kuribayashi and Hideyuki Kimura, Automorphism groups of compact Riemann surfaces of genus five, J. Algebra 134 (1990), no. 1, 80–103. MR 1068416, DOI 10.1016/0021-8693(90)90212-7
Izumi Kuribayashi and Akikazu Kuribayashi, Automorphism groups of compact Riemann surfaces of genera three and four, J. Pure Appl. Algebra 65 (1990), no. 3, 277–292. MR 1072285, DOI 10.1016/0022-4049(90)90107-S
W. B. R. Lickorish, A finite set of generators for the homeotopy group of a $2$-manifold, Proc. Cambridge Philos. Soc. 60 (1964), 769–778. MR 171269, DOI 10.1017/s030500410003824x
Aleksander Malnič, Roman Nedela, and Martin Škoviera, Lifting graph automorphisms by voltage assignments, European J. Combin. 21 (2000), no. 7, 927–947. MR 1787907, DOI 10.1006/eujc.2000.0390
William S. Massey, A basic course in algebraic topology, Graduate Texts in Mathematics, vol. 127, Springer-Verlag, New York, 1991. MR 1095046, DOI 10.1007/978-1-4939-9063-4
Atsuhiro Nakamoto, Irreducible quadrangulations of the Klein bottle, Yokohama Math. J. 43 (1995), no. 2, 125–139. MR 1414189
Atsuhiro Nakamoto, Diagonal transformations in quadrangulations of surfaces, J. Graph Theory 21 (1996), no. 3, 289–299. MR 1374901, DOI 10.1002/(SICI)1097-0118(199603)21:3<289::AID-JGT3>3.3.CO;2-2
Atsuhiro Nakamoto, Irreducible quadrangulations of the torus, J. Combin. Theory Ser. B 67 (1996), no. 2, 183–201. MR 1399674, DOI 10.1006/jctb.1996.0040
Atsuhiro Nakamoto, Generating quadrangulations of surfaces with minimum degree at least $3$, J. Graph Theory 30 (1999), no. 3, 223–234. MR 1671172, DOI 10.1002/(SICI)1097-0118(199903)30:3<223::AID-JGT7>3.3.CO;2-D
D. Pellicer and A. I. Weiss, Uniform maps on surfaces of non-negative Euler characteristic, Symmetry Culture Science 22 (2011), no. 1-2, 159–196.
Brigitte Servatius and Peter R. Christopher, Construction of self-dual graphs, Amer. Math. Monthly 99 (1992), no. 2, 153–158. MR 1144356, DOI 10.2307/2324184
Brigitte Servatius and Herman Servatius, Self-dual maps on the sphere, Discrete Math. 134 (1994), no. 1-3, 139–150. Algebraic and topological methods in graph theory (Lake Bled, 1991). MR 1303403, DOI 10.1016/0012-365X(93)E0069-G
Brigitte Servatius and Herman Servatius, The $24$ symmetry pairings of self-dual maps on the sphere, Discrete Math. 140 (1995), no. 1-3, 167–183. MR 1333718, DOI 10.1016/0012-365X(94)00293-R
Brigitte Servatius and Herman Servatius, Self-dual graphs, Discrete Math. 149 (1996), no. 1-3, 223–232. MR 1375109, DOI 10.1016/0012-365X(94)00351-I
Brigitte Servatius and Herman Servatius, Symmetry, automorphisms, and self-duality of infinite planar graphs and tilings, International Scientific Conference on Mathematics. Proceedings (Žilina, 1998) Univ. Žilina, Žilina, 1998, pp. 83–116. MR 1739919
Jozef Širáň, Coverings of graphs and maps, orthogonality, and eigenvectors, J. Algebraic Combin. 14 (2001), no. 1, 57–72. MR 1856229, DOI 10.1023/A:1011218020755
Jozef Širáň and Thomas W. Tucker, Symmetric maps, Topics in topological graph theory, Encyclopedia Math. Appl., vol. 128, Cambridge Univ. Press, Cambridge, 2009, pp. 199–224. MR 2581547
Saul Stahl, Self-dual embeddings of Cayley graphs, J. Combin. Theory Ser. B 27 (1979), no. 1, 92–107. MR 539080, DOI 10.1016/0095-8956(79)90072-8
John Stillwell, Classical topology and combinatorial group theory, 2nd ed., Graduate Texts in Mathematics, vol. 72, Springer-Verlag, New York, 1993. MR 1211642, DOI 10.1007/978-1-4612-4372-4
David B. Surowski, Lifting map automorphisms and Macbeath’s theorem, J. Combin. Theory Ser. B 50 (1990), no. 2, 135–149. MR 1081218, DOI 10.1016/0095-8956(90)90069-C
Carsten Thomassen, Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface, Trans. Amer. Math. Soc. 323 (1991), no. 2, 605–635. MR 1040045, DOI 10.1090/S0002-9947-1991-1040045-3
W. Thurston, The geometry and topology of 3-manifolds, Tech. report, Princeton University, 1980, Chapter 13; electronic version available at http://library.msri.org/books/gt3m.
Thomas W. Tucker, Finite groups acting on surfaces and the genus of a group, J. Combin. Theory Ser. B 34 (1983), no. 1, 82–98. MR 701174, DOI 10.1016/0095-8956(83)90009-6
Arthur T. White, Orientable imbeddings of Cayley graphs, Duke Math. J. 41 (1974), 353–371. MR 363975
Hassler Whitney, 2-Isomorphic Graphs, Amer. J. Math. 55 (1933), no. 1-4, 245–254. MR 1506961, DOI 10.2307/2371127
J. C. Wierman, Construction of infinite self-dual graphs, Proceedings of the 5th Hawaii International Conference on Statistics, Mathematics and Related Fields, 2006.
Thomas Zaslavsky, Biased graphs. I. Bias, balance, and gains, J. Combin. Theory Ser. B 47 (1989), no. 1, 32–52. MR 1007712, DOI 10.1016/0095-8956(89)90063-4
Robert M. Ziff and Christian R. Scullard, Exact bond percolation thresholds in two dimensions, J. Phys. A 39 (2006), no. 49, 15083–15090. MR 2277091, DOI 10.1088/0305-4470/39/49/003