Locally finite vertex-rotary maps and coset graphs with finite valency and finite edge multiplicity

A well-known theorem of Sabidussi shows that a simple G-arc-transitive graph can be represented as a coset graph for the group G. This pivotal result is the standard way to turn problems about simple arc-transitive graphs into questions about groups. In this paper, the Sabidussi representation is extended to arc-transitive, not necessarily simple graphs which satisfy a local-finiteness condition: namely graphs with finite valency and finite edge-multiplicity. The construction yields a G-arc-transitive coset graph Cos(G,H,J), where H,J are stabilisers in G of a vertex and incident edge, respectively. A first major application is presented concerning arc-transitive maps on surfaces: given a group G=〈a,z〉 with |z|=2 and |a| finite, the coset graph Cos(G,〈a〉,〈z〉) is shown, under suitable finiteness assumptions, to have exactly two different arc-transitive embeddings as a G-arc-transitive map (V,E,F) (with V,E,F the sets of vertices, edges and faces, respectively), namely, a G-rotary map if |az| is finite, and a G-bi-rotary map if |zz^a| is finite. The G-rotary map can be represented as a coset geometry for G, extending the notion of a coset graph. However the G-bi-rotary map does not have such a representation, and the face boundary cycles must be specified in addition to incidences between faces and edges. In addition a coset geometry construction is given of a flag-regular map (V,E,F) for non necessarily simple graphs. For all of these constructions it is proved that the face boundary cycles are simple cycles precisely when the given group acts faithfully on V∪F. Illustrative examples are given for graphs related to the n-dimensional hypercubes and the Petersen graph.