Coxeter Groups, Artin Groups, Buildings

The basic notion in this chapter is a Coxeter system \((W,S)\) . There are three related types of groups that depend on this notion: Coxeter groups, Artin groups, and chamber-transitive automorphism groups of buildings. In each case the group acts on an associated polyhedron. In the case of a Coxeter system the polyhedron is called the “Davis–Moussong complex;” in case of an Artin group it is the “Deligne complex;” and in the case of a building it is the “standard realization.” The action of any one of these groups has a strict fundamental domain called the “standard fundamental chamber.” It depends only on the Coxeter system. In each case we get a system of groups with the structure of a complex of groups. The fact that the fundamental domain is strict means that we have a simple complex of groups. The Davis–Moussong complex and the standard realization of a building are both \(\operatorname {CAT}(0)\) , hence, contractible. For a general Artin group, it is an open question if its Deligne complex is contractible. For an Artin group, there is a related polyhedron called the “Salvetti complex.” The Artin group is the fundamental group of the quotient of the Salvetti complex by the associated Coxeter group.