Finiteness and CAT(0) properties of diagram groups

Abstract

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 $\text{F}{}_{\mathrm{\infty }}$.