Finiteness and CAT(0) properties of diagram groups

Any diagram group over a finite semigroup presentation acts properly, freely, and cellularly by isometrices on a proper CAT(0) cubical complex. The existence of a proper, cellular action by isometries on a CAT(0) cubical complex has powerful consequences for the acting group G. One gets, for example, a proof that G satisfies the Baum–Connes conjecture. Any diagram group over a finite presentation of a finite semigroup is of type F ∞ .