Finiteness and CAT(0) properties of diagram groups.
The author proves that any diagram group over a finite semigroup presentation acts properly, freely, and cellularly by isometries on a proper \(\text{CAT}(0)\) cubical complex. As a consequence, he concludes that the group satisfies the Baum-Connes conjecture and is of type \(F_\infty\). (We recall the idea of a diagram group here [see \textit{V. Guba} and \textit{M. Sapir}, Mem. Am. Math. Soc. 620 (1997; Zbl 0930.20033)]: A diagram group is defined in terms of a semigroup presentation \(P\) and a (positive) base word \(w\) in the alphabet of \(P\). The diagram group based on (\(P,w)\) is a ``fundamental group'' based on ``loops'' \(w=v_0\to v_1\to\cdots\to v_n=w\) from \(w\) to itself. Each \(v_i\) is a (positive) word in the alphabet of \(P\), and each \(v_{i+1}\) arises from \(v_i\) by means of a defining relation \(x=y\) from \(P\) whereby a subword \(x\) of \(v_i\) is replaced by the word \(y\) to form \(v_{i+1}\). After one has defined homotopies of loops appropriately, the homotopy classes of loops form a group under concatenation. This group is called a diagram group. As an important example, Richard Thompson's group \(F\) is the diagram group based on the presentation \(\langle x\mid x=x^2\rangle\) with base word \(w=x\).)
scientific article; zbMATH DE number 1965066