University of Chicago
Department of Mathematics

(joint with Victor Guba) Diagram groups are the 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 by a result of Farley. We show that these universal covers are also 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 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.