Modelling biplanes on surfaces

A biplane is a geometry corresponding to a symmetric k2+1,k,2 block design. Nontrivial biplanes are known to exist only for k=3,4,5,6,9,11 and 13. Group difference set constructions exist for the unique biplanes having k=3,4, and 5; for all three biplanes having k=6; and for one of the four biplanes having k=9. We find models for these seven biplanes, using Cayley graph imbeddings on closed 2-manifolds.