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Automorphism groups of maps, hypermaps and dessins. (English) Zbl 1441.20003

Summary: A detailed proof is given of a theorem describing the centraliser of a transitive permutation group, with applications to automorphism groups of objects in various categories of maps, hypermaps, dessins, polytopes and covering spaces, where the automorphism group of any object is the centraliser of its monodromy group. An alternative form of the theorem, valid for finite objects, is discussed, with counterexamples based on Baumslag-Solitar groups to show how it can fail in the infinite case. The automorphism groups of objects with primitive monodromy groups are described, as are those of non-connected objects.

MSC:

20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
20B27 Infinite automorphism groups
05C10 Planar graphs; geometric and topological aspects of graph theory
14H57 Dessins d’enfants theory
52B15 Symmetry properties of polytopes
57M10 Covering spaces and low-dimensional topology
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References:

[1] G. Baumslag and D. Solitar, Some two-generator one-relator non-Hopfian groups,Bull. Amer. Math. Soc.68(1962), 199-201, doi:10.1090/s0002-9904-1962-10745-9. · Zbl 0108.02702
[2] G. V. Bely˘ı, On Galois extensions of a maximal cyclotomic field,Mathematics of the USSRIzvestiya14(1980), 247-256, doi:10.1070/im1980v014n02abeh001096.
[3] P. J. Cameron, P. M. Neumann and D. N. Teague, On the degrees of primitive permutation groups,Math. Z.180(1982), 141-150, doi:10.1007/bf01318900. · Zbl 0471.20002
[4] J. D. Dixon, The probability of generating the symmetric group,Math. Z.110(1969), 199-205, doi:10.1007/bf01110210. · Zbl 0176.29901
[5] R. Frucht, Herstellung von Graphen mit vorgegebener abstrakter Gruppe,Compositio Math.6 (1939), 239-250,http://www.numdam.org/item?id=CM_1939__6__239_0. · Zbl 0020.07804
[6] E. Girondo and G. Gonz´alez-Diez,Introduction to compact Riemann surfaces and dessins d’enfants, volume 79 ofLondon Mathematical Society Student Texts, Cambridge University Press, Cambridge, 2012, doi:10.1017/cbo9781139048910. · Zbl 1253.30001
[7] A. Grothendieck, Esquisse d’un programme (1984), in:Geometric Galois actions, 1, Cambridge Univ. Press, Cambridge, volume 242 ofLondon Math. Soc. Lecture Note Ser., pp. 5-47, 1997, with an English translation on pp. 243-283,https://webusers.imj-prg.fr/ ˜leila.schneps/grothendieckcircle/.
[8] G. A. Jones, Exotic behaviour of infinite hypermaps,Ars Math. Contemp.1(2008), 51-65, doi:10.26493/1855-3974.24.035. · Zbl 1167.05001
[9] G. A. Jones, Combinatorial categories and permutation groups,Ars Math. Contemp.10(2016), 237-254, doi:10.26493/1855-3974.545.fd5. · Zbl 1395.20002
[10] G. A. Jones, Realisation of groups as automorphism groups in categories, 2018,https:// arxiv.org/abs/1807.00547.
[11] G. A. Jones and D. Singerman, Theory of maps on orientable surfaces,Proc. London Math. Soc. (3)37(1978), 273-307, doi:10.1112/plms/s3-37.2.273. · Zbl 0391.05024
[12] G. A. Jones and J. Wolfart,Dessins d’enfants on Riemann surfaces, Springer Monographs in Mathematics, Springer, Cham, 2016, doi:10.1007/978-3-319-24711-3. · Zbl 1401.14002
[13] S. K. Lando and A. K. Zvonkin,Graphs on surfaces and their applications, volume 141 ofEncyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2004, doi:10.1007/ 978-3-540-38361-1, with an appendix by Don B. Zagier, Low-Dimensional Topology, II. · Zbl 1040.05001
[14] A. Malcev, On isomorphic matrix representations of infinite groups,Rec. Math. [Mat. Sbornik] N.S.8 (50)(1940), 405-422. · JFM 66.0088.03
[15] W. S. Massey,Algebraic topology: An introduction, Harcourt, Brace & World, Inc., New York, 1967. · Zbl 0153.24901
[16] P. McMullen and E. Schulte,Abstract regular polytopes, volume 92 ofEncyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2002, doi:10.1017/ cbo9780511546686. · Zbl 1039.52011
[17] J. R. Munkres,Topology, Prentice Hall, Inc., Upper Saddle River, NJ, 2000, second edition of [MR0464128].
[18] H. Neumann,Varieties of Groups, Springer-Verlag, Berlin - Heidelberg, 1st edition, 1967, doi: 10.1007/978-3-642-88599-0. · Zbl 0251.20001
[19] C. E. Praeger and C. Schneider,Permutation groups and Cartesian decompositions, volume 449 ofLondon Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2018, doi:10.1017/9781139194006. · Zbl 1428.20002
[20] G. Sabidussi, Graphs with given infinite group,Monatsh. Math.64(1960), 64-67, doi:10.1007/ bf01319053. · Zbl 0097.38904
[21] T.
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