Table of contents

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 containing all possible labels of edges, a set of relations defining all possible labels of cells, and a word over 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 (which corresponds to the simplest set of relations ); 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 contains soluble subgroups of any derived length. We study also distortion of subgroups in diagram groups, including the group . It turns out that the centralizers of elements and Abelian subgroups in diagram groups are always embedded without distortion. But the group contains distorted soluble subgroups.

The paper is devoted to the proof of the global solubility with respect to time and the data of the multidimensional equations of motion of a compressible viscous barotropic fluid in the Burgers approximation (in the absence of pressure). On the basis of the techniques of Orlicz spaces new estimates for the density are obtained and the existence of weak solutions of the problem in a bounded domain with no-slip boundary conditions is established. The stress-strain constitutive relation adopted in this case is strongly non-linear, that is, the viscosity coefficients are rapidly increasing functions of invariants of the deformation velocity tensor. It is also proved that each solution in the above class is a suitable weak solution which satisfies the energy identity and the mass conservation law.

In [1] and [2] the functions of rank growth were independently introduced and investigated for subgroups of a finitely generated free group. In the present paper the concept of growth of rank is extended to subgroups of an arbitrary finitely generated group G, and the dependence of the asymptotic behaviour of the above functions on the choice of a finite generating set in G is studied. For a broad class of groups (which includes, in particular, the free polynilpotent groups) estimates for the growth of rank for subgroups are obtained that generalize the wellknown Baumslag-Eidel'kind result on finitely generated normal subgroups. Some problems related to the realization of arbitrary functions as functions of rank growth for subgroups of soluble groups are treated.

Results on the values of the ordinary Radon transform in various function spaces are extended to the case of the matrix Radon transform.

An elliptic theory is constructed for operators acting in subspaces defined in terms of even pseudodifferential projections. Index formulae are obtained for operators on compact manifolds without boundary and for general boundary-value problems. A connection with Gilkey's theory of η-invariants is established.