2019/01/17 by Bottinelli, Rémi, de Peralta, Laura Grave, Kolpakov, Alexander
#05C30 #20E07 #52C20 #52C22 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1901.05710
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).