Locally Finite Vertex-Rotary Maps and Coset Graphs with Finite Valency and Finite Edge Multiplicity

Abstract

It is well-known that a simple $G$-arc-transitive graph can be represented as a coset graph for the group $G$. This representation is extended to a construction of $G$-arc-transitive coset graphs $\Cos(G,H,J)$ with finite valency and finite edge-multiplicity, where $H,J$ are stabilisers in $G$ of a vertex and incident edge, respectively. Given a group $G=ła,z\r$ with $|z|=2$ and $|a|$ finite, the coset graph $\Cos(G,ła\r,łz\r)$ is shown, under suitable finiteness assumptions, to have exactly two different arc-transitive embeddings as a $G$-arc-transitive map $(V,E,F)$, namely, a {\it $G$-rotary} map if $|az|$ is finite, and a {\it $G$-bi-rotary} map if $|z{z}^a|$ 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. We also give a coset geometry construction of a flag-regular map $(V,E,F)$. In all of these constructions we prove that the face boundary cycles are regular cycles which are simple cycles precisely when the given group acts faithfully on $V\cup F$.