| Speaker: Jozef Siran Affiliation: Slovak University of Technology Time: 14:00 Tuesday, 19 February, 2019 Location: 303-257 |
| A map, that is, a cellularly embedded graph on a surface, is regular if its automorphism group acts regularly on mutually incident vertex-edge-face triples. Regular maps exhibit the highest "level of symmetry" among all maps. Some regular maps admit, in addition "external symmetries", which are not automorphisms but make the maps even more symmetric. For example, a regular map that is self-dual or self-Petrie-dual belongs to such a category, so that one type of eternal symmetries are the two self-dualities. Another way of introducing "external symmetries" is not as easy to describe and we will do it only for regular maps on orientable surfaces to keep things simple. Suppose that one re-embeds a graph underlying a regular map of valency d on an orientable surface in such a way that, for every vertex, instead of the local cyclic ordering of edges emanating from the vertex we take the j-th power of this ordering for some fixed j coprime to d. This may be called taking the j-th rotational power of the map. If the new map happens to be isomorphic to the original one, we call j an exponent of the map. The highest level of such external "rotational power symmetry" occurs if all units mod d are exponents; such maps are known as kaleidoscopic. In general, external symmetries of a regular map are compositions of self-dualities and rotational powers corresponding to exponents. Thus, the ultimate level of symmetry is achieved by regular maps that are both kaleidoscopic as well as self-dual and self-Petrie-dual; these deserve to be called super-symmetric. In our talk we will survey results on the existence of regular maps with given valency and face length that are kaleidoscopic, or self-dual and self-Petrie-dual, or super-symmetric. The corresponding constructions are often tricky and we will have a glance at some of the underpinning methods. |