Diagram groups and directed 2-complexes: Homotopy and homology

Abstract

We show that diagram groups can be viewed as fundamental groups of spaces of positive paths on directed 2-complexes (these spaces of paths turn out to be classifying spaces). Thus diagram groups are analogs of second homotopy groups, although diagram groups are as a rule non-Abelian. Part of the paper is a review of the previous results from this point of view. In particular, we show that the so-called rigidity of the R. Thompson's group F and some other groups is similar to the flat torus theorem. We find several finitely presented diagram groups (even of type F ∞ ) each of which contains all countable diagram groups. We show how to compute minimal presentations and homology groups of a large class of diagram groups. We show that the Poincaré series of these groups are rational functions. We prove that all integer homology groups of all diagram groups are free Abelian.

Hyperbolic diagram groups are free

“…It partially answers [GS06a,Problem 9.15] by extending [GS06a,theorem 9.14] to any diagram group. As an immediate consequence, hyperbolic diagram groups turn out to be free, answering a weaker question appearing in [S07, Problem 6.1]:…”

“…The definition of diagram groups was first given by Meakin and Sapir, with the first results found by their student Kilibarda in her thesis [Kil94]. Although it was proved that diagram groups define a large class of groups with strong properties [GS97,GS99,GS06a,GS06b], very little is known on their geometric properties. Nevertheless, an important property due to Farley [Far00] is that they act freely on a CAT(0) cube complex.…”

“…In fact, Guba and Sapir [GS06a] gave a positive answer to this problem for a large class of diagram groups, namely for diagram groups over a complete directed 2-complex. Using a cohomological argument, they actually proved much more: such a diagram group is free if and only if it does not contain any subgroup isomorphic to Z 2 .…”

Universal diagram groups with identical Poincaré series

“…These groups have been studied comprehensively by V. Guba and M. Sapir in the monograph [4], and the papers [6], [7]. Other interesting work about these groups can be found in [1], [2], [3], [5], [9].…”

“…There is a subclass of the class of diagram groups which is of interest, namely those which are universal. These groups have the property that they contain a copy of every countable diagram group (see [6], §5).…”