Complete bipartite maps, factorisable groups and generalised Fermat curves

This paper is based on a talk given at the Com2MaC Conference on Applications of Group Theory to Combinatorics, July 2007. It is a brief survey of recent progress on the combinatorial problem of classifying the orientably regular embeddings of complete bipartite graphs. The motivation for this problem comes from two main areas, topological graph theory and arithmetic algebraic geometry, while the techniques required to solve it come from a third area, finite group theory-specifically the theories of factorisable groups and of finite solvable groups.