Foundations of translation planes (Q2719285)

The book under review fills a gap after the books by \textit{T. G. Ostrom} [Finite translation planes (Springer) (1970; Zbl 0205.49901)] and \textit{H. Lüneburg} [Translation planes (Springer) (1980; Zbl 0446.51003)] and provides a self-contained and rather complete picture of the present state of the art. The authors are well known leading experts on translation planes and they are preparing ``An Atlas of Planes and Processes'' on the same topics.NEWLINENEWLINENEWLINEA ``translation plane'' is an affine plane that admits a group of translations that acts transitively on its points. The Desarguesian planes satisfy this property and essentially any affine or projective plane is intrinsically connected to some translation plane. This is the reason why the literature of the subject is so rich: classical papers are due to \textit{L. A. Rosati}, \textit{A. Barlotti}, \textit{T. G. Ostrom}, \textit{D. R. Hughes}, \textit{D. A. Foulser}, and \textit{H. Lüneburg}. In the 1950's \textit{André} pointed out that points of a translation plane can be viewed as vectors of a vector space and lines as ``translate'' of a family of half-dimensional subspaces, the set of which forms a cover of the points.NEWLINENEWLINENEWLINEIn the text the authors present a wealth of information about methods, techniques and construction procedures (geometric, algebraic, group theoretic) on the subject, and therefore the book may serve as a reference and a guide to the literature for experienced researchers, as well as a textbook for graduate students who will find sufficient tools to begin research in this fascinating area of combinatorial geometry. The text constitutes a coherent and comprehensive approach to the foundation, construction and analysis of translation planes. It discusses in complete detail the theory of coordinatization of the related algebraic structures, tangentially transitive translation planes, the theorems of \textit{André}, \textit{Ostrom}, \textit{Hering} and \textit{Foulser}, spreads, collineation groups, and the fundamental analysis of matrix spreadsets.NEWLINENEWLINENEWLINESome of the contents are as follows: André's theory of spreads, partial spreads and translation nets, partial spreadsets, coordinatization of translation planes by quasifields or similar algebraic structures, rational Desarguesian nets, Hall systems, derivation of finite spreads, Foulser's covering theorem, Baer groups, planar groups, André systems, lifting and quasifibrations, tangentially transitive planes, maximal partial spreads, and the Foulser-Johnson \(SL(2,q)\)-theorem. In addition the text is equipped with eight appendices to support the essential background for students.NEWLINENEWLINENEWLINEThe book is written in a pleasant style and the material is arranged in masterly fashion. The exposition is very clear, rigorous and detailed, and the amount of material provided is remarkably large. This is an admirable book, highly recommended.