Triangle group representations and their applications to graphs and maps Original Research Article

The aim of this survey article is to draw the attention of the combinatorial community to representations of groups and their applications in the theory of graphs and maps. Specifically, we shall be interested in representations of the triangle groups 〈y,z|ym=zn=(yz)2=1〉 in special linear groups. Applications will include constructions of highly symmetrical finite maps of arbitrarily large planar width, Hurwitz groups, vertex-transitive non-Cayley graphs, and arc-transitive graphs of given valence and given exact girth.