Spherical Buildings e Poligoni di Moufang

In this thesis we give an introduction to various approaches to the notion of Tits' buildings. Tits' buildings are introduced in three different ways. The first is the classical approach: buildings are defined as a simplicial complexes with a family of subcomplexes called apartments or Coxeter complexes glued together in a certain regular fashion. The second approach is the so-called local approach: here we forget all simplices except chambers. Chambers are viewed as vertices of particular edge-colored graphs called chamber systems. A building is viewed as a set of chambers, together with a Weyl-group-valued distance function subject to a few conditions. Finally the metric approach is obtained by taking geometric realizations of the simplicial structures of a building. We will demonstrate that these three approaches to buildings, although they are distinguished by how one views chambers, are in fact all equivalent. We give a proof of Tits' extension theorem using the local approach. This theorem plays a fundamental role in the classification of spherical building of rank at least three. We also introduce the Moufang condition. This is a strong geometric homogeneity condition for buildings. Using Tits' extension theorem it is shown that every thick irreducible building of rank at least three is Moufang. Finally an overview to the classification of thick irreducible spherical buildings is given.