Cartwright et al. consider actions on buildings that are simply transitive on vertices. Each such action produces a bijection (or modified duality) between the points "x" and lines "x"* in the flag complex of a finite projective plane and a collection of oriented triangles of points ("x","y","z"), invariant under cyclic permutation, such that "x" lies on "z"*, "y" lies on "x"* and "z" lies on "y"* and any two points uniquely determine the third. The groups produced have generators "x", labelled by points, and relations "xyz" = 1 for each triangle. Generically this construction will not correspond to an action on a classical affine building.