On subgroups of R. Thompson's group $ F$ and other diagram groups

Abstract

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.

Random subgroups of Thompson’s group $F$

“…One important subgroup of F is the restricted wreath product Z Z. Guba and Sapir [13] proved a dichotomy concerning subgroups of F : any subgroup of F is either free abelian or contains a subgroup isomorphic to Z Z. A representative example of a subgroup of F isomorphic to Z Z is easily seen to be generated by the elements x 0 and y = x 1 x 2 x −2…”

Hyperbolic diagram groups are free

“…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.…”

Thompson’s group is distorted in the Thompson–Stein groups

“…Burillo, Cleary and Stein [Burillo et al 2001] showed that F(n) is quasi-isometrically embedded into F(m) for all n, m ∈ ‫,}1{−ގ‬ and along with Taback, that F is quasi-isometrically embedded in Thompson's group T [Burillo et al 2009]. Different methods have been used to show that F n × ‫ޚ‬ m is quasi-isometrically embedded in F for all m, n ∈ ‫ގ‬ [Burillo 1999;Cleary and Taback 2003;Guba and Sapir 1999;Guba and Sapir 1997]. Since the development of the main theorem of this paper, Burillo and Cleary [2010] have used similar methods as those described here to prove that the canonical embeddings of Thompson's groups F and V are also distorted in the higher dimensional Thompson's group nV .…”