3-extremal holomorphic maps and the symmetrised bidisc

We analyze the 3-extremal holomorphic maps from the unit disc D to the symmetrized bidisc G=def{(z+w,zw):z,w∈D} with a view to the complex geometry and function theory of G. These are the maps whose restriction to any triple of distinct points in D yields interpolation data that are only just solvable. We find a large class of such maps; they are rational of degree at most 4. It is shown that there are two qualitatively different classes of rational G-inner functions of degree at most 4, to be called aligned and caddywhompus functions; the distinction relates to the cyclic ordering of certain associated points on the unit circle. The aligned ones are 3-extremal. We describe a method for the construction of aligned rational G-inner functions; with the aid of this method we reduce the solution of a 3-point interpolation problem for aligned holomorphic maps from D to G to a collection of classical Nevanlinna–Pick problems with mixed interior and boundary interpolation nodes. Proofs depend on a form of duality for G.