is said to be of extra-large type if for all generators . Artin–Tits groups of extra-large type are eligible for small cancellation theory. As an application, Artin–Tits groups of extra-large type are torsion-free and have solvable conjugacy problem (Kenneth Appel and Paul Schupp). Artin–Tits groups of extra-large type are biautomatic (David Peifer). Artin groups of large type are shortlex automatic with regular geodesics (Derek Holt and Sarah Rees). Other types Many other families of Artin–Tits groups have been identified and investigated. Here we mention two of them. An Artin–Tits group is said to be of FC type ("flag complex") if, for every subset of such that for all in , the group is of spherical type. Such groups act cocompactly on a CAT(0) cubical complex, and, as a consequence, one can find a rational normal form for their elements and deduce a solution to the word problem (Joe Altobelli and Charney ). An alternative normal form is provided by multifraction reduction, which gives a unique expression by an irreducible multifraction directly extending the expression by an irreducible fraction in the spherical case (Dehornoy). An Artin–Tits group is said to be of affine type if the associated Coxeter group is affine. They correspond to the extended Dynkin diagrams of the four infinite families for , , for , and for , and of the five sporadic types , , , , and . Affine Artin–Tits groups are of Euclidean type: the associated Coxeter group acts geometrically on a Euclidean space. As a consequence, their center is trivial, and their word problem is decidable (Jon McCammond and Robert Sulway ). In 2019, a proof of the conjecture was announced for all affine Artin–Tits groups (Mario Salvetti and Giovanni Paolini). See also Free partially commutative monoid Artinian group (an unrelated notion) Non-commutative cryptography Elementary abelian group References Further reading Braid groups