Klein's Riemann Surface of Genus 3 and Regular Imbeddings of Finite Projective Planes

The simple group F of order 168 which first occurred in the work of Galois around 1830 reappeared in two geometric contexts later in the 19th century. It arose in Felix Klein's investigation in 1877 of a Riemann surface of genus 3 which admits F as its group of conformal homeomorphisms. This is the least genus for which Hurwitz's 84(g—1) bound is attained. (See [6] for an interesting historical account of Klein's surface.) It also occurred as the collineation group of the projective plane of order 2 which first appeared in Fano's work in 1894. In this note we point out a close connection between these two geometric objects. Fano's plane is an example of a hypergraph and we can consider hypergraph imbeddings in the same way as we consider graph imbeddings. As with the latter theory it is possible to consider the surface in which the hypergraph imbeds as a Riemann surface so that both a Riemannian metric and angles can be defined. We can then imbed the hypergraph so that the edges are geodesic segments and each vertex is divided into equal angles by the edges emanating from it. When one considers imbeddings of highly symmetric graphs or hypergraphs it is natural to try and choose as highly symmetric an imbedding as possible. In particular we would like the graph or hypergraph to imbed as a regular map or hypermap, that is the automorphism group or the map (resp. hypermap) should act transitively on the darts (resp. bits)-see §3 and §4. These automorphisms can then be realized as conformal homeomorphisms of the Riemann surface. In this note we show that the Fano plane can be regularly imbedded in precisely two surfaces. One imbedding which is already known [14] is on a torus and is closely related to the toroidal imbedding of the complete graph K7. The other imbedding is on Klein's Riemann surface of genus 3. This begs the question as to regular imbeddings of other projective planes. Using a result of Higman and McLaughlin we show that the only other Desarguesian projective plane that can be regularly imbedded is the projective plane of order 8. Again this can be regularly imbedded on two orientable surfaces, their genera being 220 and 252. If we drop the Desarguesian condition then it can be shown that no other planes of order less than or equal to 3600 can be regularly imbedded in an orientable surface. An analogous question for graphs concerns regular imbeddings of the complete graph Kn in an orientable surface. It is known that such imbeddings exist if and only if n is a prime power [1, 8] so that an infinite number of examples are known in this case.