Conjugacy classes of reflections of maps

Gareth Jones
University of Southampton

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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.

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