# ALGEBRAIC COMBINATORICS

Telescopic groups and symmetries of combinatorial maps
Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 483-508.

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).

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.101
Classification: 20E07,  52C20,  52C22,  05C30
Keywords: automorphism, free product, free group, hypermap, ribbon graph, symmetry
Bottinelli, Rémi 1; Grave de Peralta, Laura 1; Kolpakov, Alexander 1

1 Institut de Mathématiques Université de Neuchâtel Rue Emile-Argand 11 CH-2000 Neuchâtel, Switzerland
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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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