In the present paper, we show that many combinatorial and topological objects, such as maps, hypermaps, three-dimensional pavings, constellations and branched coverings of the two-sphere admit any given finite automorphism group. This enhances the already known results by Frucht, Cori–Machì, Širáň–Škoviera, and other authors. We also provide a more universal technique for showing that “any finite automorphism group is possible”, that is applicable to wider classes or, in contrast, to more particular sub-classes of said combinatorial and geometric objects. Finally, we show that any given finite automorphism group can be realised by sufficiently many non-isomorphic such entities (super-exponentially many with respect to a certain combinatorial complexity measure).
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Keywords: automorphism, free product, free group, hypermap, ribbon graph, symmetry
Bottinelli, Rémi 1; Grave de Peralta, Laura 1; Kolpakov, Alexander 1
CC-BY 4.0
@article{ALCO_2020__3_2_483_0,
author = {Bottinelli, R\'emi and Grave de Peralta, Laura and Kolpakov, Alexander},
title = {Telescopic groups and symmetries of combinatorial maps},
journal = {Algebraic Combinatorics},
pages = {483--508},
publisher = {MathOA foundation},
volume = {3},
number = {2},
year = {2020},
doi = {10.5802/alco.101},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.101/}
}
TY - JOUR AU - Bottinelli, Rémi AU - Grave de Peralta, Laura AU - Kolpakov, Alexander TI - Telescopic groups and symmetries of combinatorial maps JO - Algebraic Combinatorics PY - 2020 SP - 483 EP - 508 VL - 3 IS - 2 PB - MathOA foundation UR - https://alco.centre-mersenne.org/articles/10.5802/alco.101/ DO - 10.5802/alco.101 LA - en ID - ALCO_2020__3_2_483_0 ER -
%0 Journal Article %A Bottinelli, Rémi %A Grave de Peralta, Laura %A Kolpakov, Alexander %T Telescopic groups and symmetries of combinatorial maps %J Algebraic Combinatorics %D 2020 %P 483-508 %V 3 %N 2 %I MathOA foundation %U https://alco.centre-mersenne.org/articles/10.5802/alco.101/ %R 10.5802/alco.101 %G en %F ALCO_2020__3_2_483_0
Bottinelli, Rémi; Grave de Peralta, Laura; Kolpakov, Alexander. Telescopic groups and symmetries of combinatorial maps. Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 483-508. doi : 10.5802/alco.101. https://alco.centre-mersenne.org/articles/10.5802/alco.101/
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