Geometry of sporadic groups. II: Representations and amalgams (Q2781474)

This is the second volume of the two-volume series which provides a complete self-contained proof of the classification of flag-transitive Petersen and tilde geometries associated with sporadic simple groups. The final result of the classification was announced by the authors in [Bull. Am. Math. Soc., New Ser. 31, No. 2, 173-184 (1994; Zbl 0817.20014)]. In the first volume [``Geometry of sporadic groups. I: Petersen and tilde geometries'' (Encyclopedia of Mathematics and Its Applications. 76. Cambridge University Press, Cambridge) (1999; Zbl 0933.51006)], \textit{A. A. Ivanov} gave a construction and the calculation of the universal cover for each of the known flag-transitive Petersen and tilde geometries. NEWLINENEWLINENEWLINEIn this second volume, it is proved that each flag-transitive Petersen and tilde geometry of rank at least 3 is isomorphic to a geometry described in the first volume. Part I (Representations) of this volume contains a study of the representations of the geometries under consideration in \(GF(2)\)-vector spaces as well as in some non-abelian groups. The central Part II (Amalgams) is the classification of the amalgams of maximal parabolics associated with flag-transitive actions on Petersen and tilde geometry. This classification is based on the method of group amalgams, the most promising tool in modern finite group theory. NEWLINENEWLINENEWLINEIn the introductory Chapter I, the main notions concerning diagram geometries are recalled and the fundamental principle, which relates the universal cover of a geometry and the universal completion of the amalgams of maximal parabolics in a flag-transitive automorphism group of this geometry, is proved. NEWLINENEWLINENEWLINEIn Chapter 2, some techniques for calculating representations of geometries of \(GF(2)\)-type, i.e., with three points on line is given. NEWLINENEWLINENEWLINEIn Chapter 3, the representations of the classical geometries of \(GF(2)\)-type and of the tilde geometries of symplectic type are studied. NEWLINENEWLINENEWLINEIn Chapters 4 and 5, the representations of geometries associated with Mathieu groups, the Held group and Conway groups are studied. NEWLINENEWLINENEWLINEIn Chapter 6, representations of involution geometries of groups are considered. NEWLINENEWLINENEWLINEIn Chapter 7, the universal representations of 2-local parabolic geometries of large sporadics \(F_{24}'\), \(J_4\), \(BM\), and \(M\) are calculated. NEWLINENEWLINENEWLINEIn Chapter 8, the authors collect and develop some machinery for classifying the amalgams of maximal parabolics coming from flag-transitive actions on Petersen and tilde geometries. NEWLINENEWLINENEWLINEIn Chapter 9, the flag-transitive action on the derived graph of a Petersen or tilde geometry is studied. NEWLINENEWLINENEWLINEIn Chapter 10, a relatively short list of possible shapes of amalgams of maximal parabolics are obtained. NEWLINENEWLINENEWLINEIn Chapters 11 and 12, the authors classify the amalgams of maximal parabolics for Petersen and tilde geometries, respectively. It is shown, that some of the shapes from Chapter 10 are impossible, and the others lead to the actual examples. NEWLINENEWLINENEWLINEIn Chapter 13, the authors discuss two projects which lie beyond the classification of the flag-transitive Petersen and tilde geometries. They report on the latest progress in the attempt to classify the Petersen and tilde geometries when the flag-transitive assumption is dropped and discuss the problem of the classification of all amalgams of vertex- and edge-stabilizers coming from locally projective actions of groups on graphs. NEWLINENEWLINENEWLINEThis book is of great interest to researcher in finite group theory, finite geometries and algebraic combinatorics.