Maximality Properties of Stable Partial Spreads and of Shear Planes

Abstract

Shear planes are special stable planes which can be derived from stable partial spreads by a construction analogous to the construction of (punctured) dual translation planes from spreads. We give a sufficient condition for the maximality of shear planes in the category of stable planes and open embeddings. The condition is expressed in terms of the defining partial spread. Moreover, we give several examples of connected and disconnected shear planes in dimension 4, 8 and 16 which are maximal stable planes and have large automorphism groups. Most of these examples are direct limits of open subplanes of dual projective translation planes.