Conjugacy classes of reflections of maps
University of Southampton
Minisymposium: GRAPH IMBEDDINGS AND MAP SYMMETRIES
Content: I shall consider how many conjugacy classes of reflections a map can have, under various transitivity conditions. For vertex- and for face-transitive maps there are no restrictions on their number or size, whereas edge-transitive maps can have at most four classes of reflections. This upper bound is attained only by certain maps of type 3 in the Graver-Watkins taxonomy of edge-transitive maps. (These are the just-edge-transitive maps, those which are edge- but neither vertex- nor face-transitive.) Examples are constructed, using topology, covering spaces and group theory, to show that various distributions of reflections can be achieved.