A characterization of free projective planes

Abstract

In a fundamental paper Marshall Hall defines a free plane to be a projective plane which either is degenerate or is generated as follows from a 'basis configuration/ ττ0, consisting of at least two points on a line together with two isolated points. For each pair of points not already joined in π0 create a distinct line that joins them and add it to 7Γ0. In the resulting configuration, πlf consider pairs of lines that do not intersect, and for each create a distinct intersection point and add it to πu thus forming π2. Continuing, construct 7Γ3, τr4, πδ, π6, etc. adding alternately lines and points as indicated above. Then π = U*^* (with the obvious incidence relation) is a projective plane. It is bydefinition a free plane. Hall proved that a free plane contains no confined configuration, that is, no finite configuration that, like the Desargues configuration, has ^ 3 points on each line and ^ 3 lines through each point. Further, using a complicated argument, he showed that, if a finitely generated plane contains no confined configuration, it is free. It follows that any finitely generated subplane of a free plane is free. L. I. Kopejkina [2] proved, shortly after, that an arbitrary subplane of a free plane is free. (Of interest is the analogy with free groups.) An exposition of Kopejkina's theorem appears in [3]. Because it suggests a more symmetrical definition of free plane that leads to very direct proofs of the above theorems, we introduce the notion of an extension process.