Baer subplanes (Q1409628)

This is an extensive, clear and concise survey of the ideas and results that developed out of \textit{R. Baer's} 1946 seminal paper on `projectivities with fixed points on every line of the plane' [Bull. Am. Math. Soc. 52, 273--286 (1946; Zbl 0061.30810)]. The basic notions are that of a Baer subplane of a projective plane (a subplane meeting every line and every line pencil; examples are obtained from quadratic field extensions, e.g., the real plane inside the complex plane) and Baer collineations (fixing all points of a Baer subplane). The paper is divided equally among results on Baer subplanes in general projective planes, in finite planes, and in compact connected planes; the latter area is the one to which the author has contributed many results. The topics treated in each part of the paper include characterization of Baer subplanes (as special blocking sets in the general case, as subplanes whose order is the square root of that of the big plane in the finite case, as subplanes of half the dimension in the compact connected case), relationships between the structure of (sets of many) Baer subplanes and of the entire plane, applications of Baer subplanes to the construction of new planes out of given ones (derivation, Hughes planes, Hall planes), groups fixing a Baer subplane pointwise and their transitivity properties, existence (in the finite case) and non-existence (in the compact connected case) of disjoint Baer subplanes. The paper comes with a bibliography of 78 items.