On external symmetry groups of regular maps

Regular maps are embeddings of graphs or multigraphs on closed surfaces (which may be orientable or non-orientable), in which the automorphism group of the embedding acts regularly on flags. Such maps may admit external symmetries that are not automorphisms of the embedding, but correspond to combinations of well known operators that may transform the map into an isomorphic copy: duality, Petrie duality, and the ‘hole operators’, also known as ‘taking exponents’. The group generated by the external symmetries admitted by a regular map is the external symmetry group of the map. We will be interested in external symmetry groups of regular maps in the case when the map admits both the above dualities (that is, if it has trinity symmetry) and all feasible hole operators (that is, if it is kaleidoscopic). Existence of finite kaleidoscopic regular maps was conjectured for every even valency by Wilson, and proved by Archdeacon, Conder and Širáň (2010). It is well known that regular maps may be identified with quotients of extended triangle groups. In other words, these groups may be regarded as ‘universal’ for constructions of regular maps. It is therefore interesting to ask if similar ‘universal’ groups exist for kaleidoscopic regular maps with trinity symmetry. A satisfactory answer, however, is likely to be very complex, if indeed feasible at all. We demonstrate this (and other things) by a construction of an infinite family of finite kaleidoscopic regular maps with trinity symmetry, all of valency 8, such that the orders of their external symmetry groups are unbounded. Also we explicitly determine the external symmetry groups for the family of kaleidoscopic regular maps of even valency mentioned above