and the cohomology is determined (Pierre Deligne, by geometrical methods, Egbert Brieskorn and Kyoji Saito, by combinatorial methods ). A pure Artin–Tits group of spherical type can be realized as the fundamental group of the complement of a finite hyperplane arrangement in . Artin–Tits groups of spherical type are biautomatic groups (Ruth Charney). In modern terminology, an Artin–Tits group is a Garside group, meaning that is a group of fractions for the associated monoid and there exists for each element of a unique normal form that consists of a finite sequence of (copies of) elements of and their inverses ("symmetric greedy normal form") Right-angled Artin groups An Artin–Tits group is said to be right-angled if all coefficients of the Coxeter matrix are either or , i.e., all relations are commutation relations . The names (free) partially commutative group, graph group, trace group, semifree group or even locally free group are also common. For this class of Artin–Tits groups, a different labeling scheme is commonly used. Any graph on vertices labeled defines a matrix , for which if the vertices and are connected by an edge in , and otherwise. The class of right-angled Artin–Tits groups includes the free groups of finite rank, corresponding to a graph with no edges, and the finitely-generated free abelian groups, corresponding to a complete graph. Every right-angled Artin group of rank r can be constructed as HNN extension of a right-angled Artin group of rank , with the free product and direct product as the extreme cases. A generalization of this construction is called a graph product of groups. A right-angled Artin group is a special case of this product, with every vertex/operand of the graph-product being a free group of rank one (the infinite cyclic group). The word and conjugacy problems of a right-angled Artin–Tits group are decidable, the former in linear time, the group is torsion-free, and there is an explicit cellular finite (John Crisp, Eddy Godelle, and Bert Wiest). Every right-angled Artin–Tits group acts freely and cocompactly on a finite-dimensional CAT(0) cube complex, its "Salvetti complex". As an application, one can use right-angled Artin groups and their Salvetti complexes to construct groups with given finiteness properties (Mladen Bestvina and Noel Brady ) see also (Ian Leary ). Artin–Tits groups of large type An Artin–Tits group (and a Coxeter group) is said to be of large type if for all generators ; it