In this paper, we continue our study of the class of diagram groups. Simply speaking, a diagram is a labelled plane graph bounded by a pair of paths (the top path and the bottom path). To multiply two diagrams, one simply identifies the top path of one diagram with the bottom path of the other diagram, and removes pairs of "reducible" cells. Each diagram group is determined by an alphabet X, containing all possible labels of edges, a set of relationscontaining all possible labels of cells, and a word w over X -the label of the top and bottom paths of diagrams. Diagrams can be considered as 2-dimensional words, and diagram groups can be considered as 2-dimensional analogue of free groups. In our previous paper, we showed that the class of diagram groups contains many interesting groups including the famous R. Thompson group F (it corresponds to the simplest set of relations { x = x 2 }), closed under direct and free products and some other constructions. In this paper we study mainly subgroups of diagram groups. We show that not every subgroup of a diagram group is itself a diagram group (this answers a question from the previous paper). We prove that every nilpotent subgroup of a diagram group is abelian, every abelian subgroup is free, but even the Thompson group contains solvable subgroups of any degree. We also study distortion of subgroups in diagram groups, including the Thompson group. It turnes out that centralizers of elements and abelian subgroups are always undistorted, but the Thompson group contains distorted soluble subgroups.
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.
We give first examples of finitely generated groups having an intermediate, with values in (0, 1), Hilbert space compression (which is a numerical parameter measuring the distortion required to embed a metric space into Hilbert space). These groups include certain diagram groups. In particular, we show that the Hilbert space compression of Richard Thompson's group F is equal to 1/2, the Hilbert space compression of Z ≀ Z is between 1/2 and 3/4, and the Hilbert space compression of Z ≀ (Z ≀ Z) is between 0 and 1/2. In general, we find a relationship between the growth of H and the Hilbert space compression of Z ≀ H.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.