Abstract: Diagram groups are fundamental groups of spaces of positive paths of directed 2-complexes. These spaces are K(.,1) and their universal covers are CAT(0) cubical complexes. In particular, the R. Thompson group F is the diagram group of the Dunce hat. This fact allows us to find cenralizers of elements in F, solve conjugacy problem there, etc.

We show that universal covers of spaces of positive paths are themselves spaces of positive paths of the so called rooted 2-trees. Using that, we prove that all homology groups of diagram groups are free abelian, and that the diagram groups are bi-orderable. We construct many new examples of infinite dimensional FP_infty groups with rational Poincare series, show that basically "any" rational function can be the Poincare function of an FP_infty group. We also construct an FP_infty diagram group containing all countable diagram groups.

I am going to talk also about representations of diagram groups by homeomorphisms of the real line, and about rigidity results involving diagram groups. The latter results are similar in spirit to the flat torus theorem but the groups involved are more complicated.

This is joint work with Victor Guba.