Description
Book SynopsisThe Handbook of Finite Translation Planes provides a comprehensive listing of all translation planes derived from a fundamental construction technique, an explanation of the classes of translation planes using both descriptions and construction methods, and thorough sketches of the major relevant theorems.
From the methods of André to coordinate and linear algebra, the book unifies the numerous diverse approaches for analyzing finite translation planes. It pays particular attention to the processes that are used to study translation planes, including ovoid and Klein quadric projection, multiple derivation, hyper-regulus replacement, subregular lifting, conical distortion, and Hermitian sequences. In addition, the book demonstrates how the collineation group can affect the structure of the plane and what information can be obtained by imposing group theoretic conditions on the plane. The authors also examine semifield and division ring planes and introduce the geometries of two-dimensional translation planes.
As a compendium of examples, processes, construction techniques, and models, the Handbook of Finite Translation Planes equips readers with precise information for finding a particular plane. It presents the classification results for translation planes and the general outlines of their proofs, offers a full review of all recognized construction techniques for translation planes, and illustrates known examples.
Trade Review"The authors, who are the undisputed leaders in the subject, present the huge material in shorter but virtually independent chapters, each dedicated to a particular aspect, such as the connection between translation planes and quasifields... This book highly recommended for the very clear, rigorous and detailed expostion and cannot be missing in the library of any researcher in Geometry."
-Bambina Larato, Zentralblatt MATH, 2008, 1136
Table of ContentsPreface and Acknowledgments. An Overview. Translation Plane Structure Theory. Partial Spreads and Translation Nets. Partial Spreads and Generalizations. Quasifields. Derivation. Frequently Used Tools. Sharply Transitive Sets. SL(2, p) × SL(2, p)-Planes. Classical Semifields. Groups of Generalized Twisted Field Planes. Nuclear Fusion in Semifields. Cyclic Semifields. T-Cyclic GL(2, q)-Spreads. Cone Representation Theory. André Net Replacements and Ostrom-Wilke Generalizations. Foulser's ?-Planes. Regulus Lifts, Intersections over Extension Fields. Hyper-Reguli Arising from André Hyper-Reguli. Translation Planes with Large Homology Groups. Derived Generalized André Planes. The Classes of Generalized André Planes. C-System Nearfields. Subregular Spreads. Fano Configurations. Fano Configurations in Generalized André Planes. Planes with Many Elation Axes. Klein Quadric. Parallelisms. Transitive Parallelisms. Ovoids.
Known Ovoids. Simple T-Extensions of Derivable Nets. Baer Groups on Parabolic Spreads. Algebraic Lifting. Semifield Planes of Orders q4, q6. Known Classes of Semifields. Methods of Oyama-Suetake Planes. Coupled Planes. Hyper-Reguli. Subgeometry Partitions. Groups on Multiple Hyper-Reguli. Hyper-Reguli of Dimension 3. Elation-Baer Incompatibility. Hering-Ostrom Elation Theorem. Baer-Elation Theory. Spreads Admitting Unimodular Sections-Foulser-Johnson Theorem. Spreads of Order q2-Groups of Order q2. Transversal Extensions. Indicator Sets. Geometries and Partitions. Maximal Partial Spreads. Sperner Spaces. Conical Flocks. Ostrom and Flock Derivation. Transitive Skeletons. BLT-Set Examples. Many Ostrom-Derivates. Infinite Classes of Flocks. Sporadic Flocks. Hyperbolic Fibrations. Spreads with Many Homologies. Nests of Reguli. Chains. Multiple Nests. A Few Remarks on Isomorphisms. Flag-Transitive Geometries. Quartic Groups in Translation Planes. Double Transitivity. Triangle Transitive Planes. Hiramine-Johnson-Draayer Theory. Bol Planes. 2/3-Transitive Axial Groups. Doubly Transitive Ovals and Unitals. Rank 3 Affine Planes. Transitive Extensions. Higher-Dimensional Flocks. j…j-Planes. Orthogonal Spreads. Symplectic Groups-The Basics. Symplectic Flag-Transitive Spreads. Symplectic Spreads. When Is a Spread Not Symplectic? When Is a Spread Symplectic? The Translation Dual of a Semifield. Unitals in Translation Planes. Hyperbolic Unital Groups. Transitive Parabolic Groups. Doubly Transitive Hyperbolic Unital Groups. Retraction. Multiple Spread Retraction. Transitive Baer Subgeometry Partitions. Geometric and Algebraic Lifting. Quasi-Subgeometry Partitions. Hyper-Regulus Partitions. Small-Order Translation Planes. Dual Translation Planes and Their Derivates. Affine Planes with Transitive Groups. Cartesian Group Planes-Coulter-Matthews. Planes Admitting PGL(3, q). Planes of Order = 25. Real Orthogonal Groups and Lattices. Aspects of Symplectic and Orthogonal Geometry. Fundamental Results on Groups. Atlas of Planes and Processes. Bibliography. Theorems. Models. General Index.
