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

In the present paper we continue our study of an interesting class of groups, the so-called diagram groups. In simple terms, a diagram is a labelled planar graph bounded by two paths (the top and the bottom ones). Multiplication of diagrams is defined naturally: the top path of one diagram is identified with the bottom path of another diagram, and then pairs of “cancellable” cells are deleted. Each diagram group is determined by some alphabet $X$ containing all possible labels of edges, a set of relations $\mathscr R=\{u_i=v_i:i=1,2,\dots\}$ defining all possible labels of cells, and a word $w$ over $X$ that is the label of the top and bottom paths of diagrams. Diagrams may be regarded as two-dimensional words, and diagram groups as two-dimensional analogues of free groups. In our previous paper we showed that the class of diagram groups contains many interesting groups, including the famous R. Thompson's group $F$ (which corresponds to the simplest set of relations $\{x=x^2\}$); this class is closed under direct and free products and a number of other constructions. In this article we study mainly subgroups of diagram groups. We show that not every subgroup of a diagram group is itself a diagram group (an answer to 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 group $F$ contains soluble subgroups of any derived length. We study also distortion of subgroups in diagram groups, including the group $F$. It turns out that the centralizers of elements and Abelian subgroups in diagram groups are always embedded without distortion. But the group $F$ contains distorted soluble subgroups.