Diagram groups: homotopy and homology
by
Mark Sapir
Vanderbilt University
Coauthors: Victor Guba

A diagram group is the fundamental group of the space of positive paths on a directed 2-complex. All known infinite dimensional FP_\infty torsion free groups which are not finite dimensional are diagram groups (the R. Thompson group F is one of them). We compute the Poincare series of FP_\infty diagram groups (all of them turn out to be rational functions). Since the spaces of positive paths are H-spaces, there are natural Pontrjagin algebras associated with diagram groups. We construct an F_\infty diagram group that contains all countable diagram groups (in particular, it contains infinite direct and free powers of itself). Finally, we show that all diagram groups are orderable.

Date received: February 20, 2003