Points and lines. Characterizing the classical geometries.

*(English)*Zbl 1213.51001
Universitext. Berlin: Springer (ISBN 978-3-642-15626-7/pbk; 978-3-642-15627-4/ebook). xxii, 676 p. (2011).

The book under review grew from the author’s lecture notes to various courses over the years and his longstanding interest in and research into Lie incidence geometries. It is intended as a teaching book and thus is self-contained. Since the text contains complete proofs of many classical theorems scattered throughout the literature, it will be valuable to experts as a reference. Moreover, the later chapters, in particular, may be of interest due to new presentations of standard theories or some new results.

The author’s primary aim is to introduce the classical geometries of Lie type and then to characterize them by simple axioms on points and lines. The text is written very clearly and abounds with footnotes, historical and other remarks, explanations on where the author is going, forward references, all intended to give perspective to the reader. Each of the 18 chapters begins with an abstract. Almost every section has an introduction and most chapters end with a set of exercises.

The book is divided into four parts, the first of which introduces the basics. The four chapters in this part deal with graphs, concepts of geometries (defined in terms of multipartite graphs as geometries over a typeset), point-line geometries, and Teirlinck’s theory (of geometric hyperplanes and embeddings), respectively. The second part introduces the classical geometries: projective planes in chapter 5, and projective spaces (including proofs of the Veblen-Young theorem and the fundamental theorem of projective geometry), polar spaces (including the determination of non-degenerate polar spaces of polar rank at least 4) and near polygons (including Cameron’s characterization of near polygons with strongly gated quads as classical dual polar spaces) in the following three chapters.

Part III, called methodology, is where a unifying view of geometries of Lie type is presented and the tools for the characterizations in the final part are developed. Chapter 9 introduces chamber systems and buildings. Basic properties of buildings are deduced directly from the strong-gatedness of their residues, thus avoiding the use of simplicial complexes. The next chapter deals with 2-covers of chamber systems and gives a full account of J. Tits’ “local approach theorem”. In chapter 11 locally truncated diagram geometries are investigated. The author follows the sheaf-theoretic approach of A. Brouwer and A. Cohen to show that a locally truncated geometry of rank at least 4 is realizable as a truncation of a geometry belonging to a diagram. Chapter 12 considers separated systems of singular spaces, and the last chapter in this part develops Cooperstein’s theory of symplecta and parapolar spaces in the wider context of “polar families” .

Part IV of the book, containing five chapters, looks at applications to other Lie incidence geometries and features the characterizations the author is aiming for. In chapter 14 three characterizations of the classical Grassmann spaces are given. This is continued in the following chapter, which characterizes the classical strong parapolar spaces. The Cohen-Cooperstein theorem on locally connected parapolar spaces all of whose symplecta possess a constant symplectic rank 3 is extended to rank \(\geq 3\) with additional assumptions in case the rank exceeds 3. Chapter 16 deals with characterizations of strong parapolar spaces by the relation between points and certain maximal singular subspaces. The next chapter investigates the “long root geometries” and presents point-line characterizations of them. The two main results are first the identification of \(E_{6,4}\), \(E_{7,7}\), \(E_{8,1}\), metasymplectic spaces, and polar Grassmannians of lines of a non-degenerate polar space of rank at least 4 as parapolar spaces of symplectic rank at least 3 satisfying four axioms, and second the identification of \(D_{6,6}\), \(A_{5,3}\), \(E_{7,1}\), dual polar spaces of rank 3, and product geometries \(L\times P\) of a line \(L\) and a non-degenerate polar space \(P\) of rank at least 2 as strong parapolar spaces satisfying three axioms. In the final chapter of the book a peculiar pentagon property is studied. Under the assumption that Cohen’s pentagon axiom holds for point residues of a parapolar space of symplectic rank at least 3 plus three other properties on singular subspaces and symplecta, a complete list of possible geometries is obtained.

The author’s primary aim is to introduce the classical geometries of Lie type and then to characterize them by simple axioms on points and lines. The text is written very clearly and abounds with footnotes, historical and other remarks, explanations on where the author is going, forward references, all intended to give perspective to the reader. Each of the 18 chapters begins with an abstract. Almost every section has an introduction and most chapters end with a set of exercises.

The book is divided into four parts, the first of which introduces the basics. The four chapters in this part deal with graphs, concepts of geometries (defined in terms of multipartite graphs as geometries over a typeset), point-line geometries, and Teirlinck’s theory (of geometric hyperplanes and embeddings), respectively. The second part introduces the classical geometries: projective planes in chapter 5, and projective spaces (including proofs of the Veblen-Young theorem and the fundamental theorem of projective geometry), polar spaces (including the determination of non-degenerate polar spaces of polar rank at least 4) and near polygons (including Cameron’s characterization of near polygons with strongly gated quads as classical dual polar spaces) in the following three chapters.

Part III, called methodology, is where a unifying view of geometries of Lie type is presented and the tools for the characterizations in the final part are developed. Chapter 9 introduces chamber systems and buildings. Basic properties of buildings are deduced directly from the strong-gatedness of their residues, thus avoiding the use of simplicial complexes. The next chapter deals with 2-covers of chamber systems and gives a full account of J. Tits’ “local approach theorem”. In chapter 11 locally truncated diagram geometries are investigated. The author follows the sheaf-theoretic approach of A. Brouwer and A. Cohen to show that a locally truncated geometry of rank at least 4 is realizable as a truncation of a geometry belonging to a diagram. Chapter 12 considers separated systems of singular spaces, and the last chapter in this part develops Cooperstein’s theory of symplecta and parapolar spaces in the wider context of “polar families” .

Part IV of the book, containing five chapters, looks at applications to other Lie incidence geometries and features the characterizations the author is aiming for. In chapter 14 three characterizations of the classical Grassmann spaces are given. This is continued in the following chapter, which characterizes the classical strong parapolar spaces. The Cohen-Cooperstein theorem on locally connected parapolar spaces all of whose symplecta possess a constant symplectic rank 3 is extended to rank \(\geq 3\) with additional assumptions in case the rank exceeds 3. Chapter 16 deals with characterizations of strong parapolar spaces by the relation between points and certain maximal singular subspaces. The next chapter investigates the “long root geometries” and presents point-line characterizations of them. The two main results are first the identification of \(E_{6,4}\), \(E_{7,7}\), \(E_{8,1}\), metasymplectic spaces, and polar Grassmannians of lines of a non-degenerate polar space of rank at least 4 as parapolar spaces of symplectic rank at least 3 satisfying four axioms, and second the identification of \(D_{6,6}\), \(A_{5,3}\), \(E_{7,1}\), dual polar spaces of rank 3, and product geometries \(L\times P\) of a line \(L\) and a non-degenerate polar space \(P\) of rank at least 2 as strong parapolar spaces satisfying three axioms. In the final chapter of the book a peculiar pentagon property is studied. Under the assumption that Cohen’s pentagon axiom holds for point residues of a parapolar space of symplectic rank at least 3 plus three other properties on singular subspaces and symplecta, a complete list of possible geometries is obtained.

Reviewer: Günter F. Steinke (Christchurch)

##### MSC:

51-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry |

51-02 | Research exposition (monographs, survey articles) pertaining to geometry |

51Axx | Linear incidence geometry |

51E24 | Buildings and the geometry of diagrams |

20E42 | Groups with a \(BN\)-pair; buildings |

51N30 | Geometry of classical groups |