## Finite geometries, buildings, and related topicsThe theory of buildings was introduced by J. Tits in order to focus on geometric and combinatorial aspects of simple groups of Lie type. Since then, the theory has blossomed into an extremely active field of mathematical research having deep connections with topics as diverse as algebraic groups, arithmetic groups, finite simple groups, and finite geometries, as well as with graph theory and other aspects of combinatorics. This volume is intended to provide an up-to-date survey of the theory of buildings with special emphasis on its interaction with related geometries. Experts in their respective fields provide coverage of such topics as the classification and construction of buildings, finite groups associated with building-like geometries, graphs and associated schemes, and more. |

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### Contents

Spheres of Radius 2 in Triangle Buildings I | 17 |

A Census of Finite Generalized Quadrangles | 29 |

Finite Geometries via Algebraic Affine Buildings | 37 |

Copyright | |

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### Common terms and phrases

3-forms affine building association scheme assume automorphisms chamber system character table circle class number classification codes common point commutative association schemes conjugacy classes conjugate connected contain a common contradiction corresponding cosets Coxeter group curves defines a linear denoted Desarguesian diagram distance-transitive graphs elements example field Finite Geometries finite simple groups Frobenius group Geom GF(q group association scheme group G Hecke algebra Hence inversive plane irreducible characters irreducible conics isomorphic J. A. Thas Lecture Notes linear flock locally Math matrix Moufang loops multiplicity-free mutually tangent order q orthogonal pair parabolic isomorphic permutation groups plane of order polar spaces polygons polynomials preprint projective plane proof quadrangles quadric rank 2 parabolics rank 3 residue resp result S. E. Payne simplicial complex singular spherical type subspace symmetric T-SCABs Thas flock Theorem Tits triangle building trilinear forms unique universal 2-cover vertex vertices W. M. Kantor