Algebraic Structure of Tensor Completions of 2-Nilpotent Torsion-Free Groups

This paper has two sections concerned with the characterization of the algebraic structures of certain types of infinite abelian groups such as compact groups, or tensor products of torsion groups. It also has a section which gives an approach to the problems of mixed abelian groups (those which are neither torsion nor torsion-free) by establishing a duality between torsion groups and those abelian groups which are as mixed as (that is, they are as far away as possible from being just a direct sum of a torsion group and a torsion-free group). There is a final section containing some miscellaneous results on torsion-free abelian groups. For the reader unfamiliar with homological algebra, we give in Section 1 a brief outline of some results and methods of that theory. Almost all our proofs will depend heavily on homological methods. Section 2 is devoted to the duality of which we have spoken. A reduced group is called co-torsion if it is always a direct summand whenever it appears as a subgroup with a torsion-free quotient group. In the same way that Pontrjagin duality reduces the problems of compact groups to those of discrete groups, the problems of co-torsion groups are reduced to those of torsion groups by a one-to-one duality between the groups of these two classes. A method is then given by which all the mixed groups with a given torsion group can be constructed from the co-torsion group dual to that torsion group. The main result of Section 3 is that an abelian group can be a compact topological group if and only if it is isomorphic to a direct product (unrestricted direct sum) of copies of finite cyclic groups, p-adic integers, the reals, and the groups Z(p-) where, for each prime p, the number of copies of Z(p-) does not exceed the number of copies of the reals. It is also proved that if G is an abelian group with the multiples n! G considered as neighborhoods of the identity, then essentially G is a direct summand of a direct product of finite cyclic groups if and only if it is complete in the metric defined by these neighborhoods, while G is a direct sum of cyclic groups if and only if it is as far from complete as possible in this metric. At the very end of Section 3 it is shown that a torsion-free group is a direct summand of a direct product of finite 366

The Steenrod algebra structures of ${H^\ast }(BG;Z/p)$ for compact Lie groups are studied. Using these, Brown-Peterson cohomology and Morava $K$-theory are computed for many concrete cases. All these cases have properties similar as torsion free Lie groups or finite groups, e.g., $B{P^{odd}}(BG) = 0$.

The Steenrod algebra structures of H ∗ ( B G ; Z / p ) {H^\ast }(BG;Z/p) for compact Lie groups are studied. Using these, Brown-Peterson cohomology and Morava K K -theory are computed for many concrete cases. All these cases have properties similar as torsion free Lie groups or finite groups, e.g., B P o d d ( B G ) = 0 B{P^{odd}}(BG) = 0 .

An investigation of the structure of the quotient algebra, with respect to a prime ideal, of the group algebra of a finitely-generated nilpotent torsion-free group. Conditions are studied under which an irreducible representation of such a group is induced.

In 1997, B. I. Plotkin introduced a concept of geometric equivalence of algebraic structures and posed a question: is it true that every nilpotent torsion-free group is geometrically equivalent to its Mal’cev’s closure? A negative answer in the form of three counterexamples was given by V. V. Bludov and B. V. Gusev in 2007. In the present paper, an infinite series of counterexamples of unbounded Hirsch rank and nilpotency degree is constructed.

Let Γ be a discrete group and p be a prime. One of the fundamental results in group cohomology is that H*(Γ, [ ]p) is a finitely generated [ ]p-algebra if Γ is a finite group [8, 24]. The purpose of this paper is to study the analogous question when Γ is no longer finite.Recall that Γ is said to have finite virtual cohomological dimension (vcd) if there exists a finite index torion-free subgroup Γ′ of Γ such that Γ′ has finite cohomological dimension over ℤ [4]. By definition vcd Γ is the cohomological dimension of Γ′. It is easy to see that the mod p cohomology ring of a finite vcd-group does not have to be a finitely generated [ ]p-algebra in general. For instance, if Γ is a countably infinite free product of ℤ's, then H1(Γ, [ ]p) is not finite dimensional over [ ]p. The three most important classes of examples of finite vcd-groups in which the mod p cohomology ring is a finitely generated [ ]p-algebra are arithmetic groups [2], mapping class groups [9, 10] and outer automorphism groups of free groups [5]. In each of these examples, the proof of finite generation involves the construction of a specific Γ-complex with appropriate finiteness conditions. These constructions should be regarded as utilizing the geometry underlying these special classes of groups. In contrast, the result we prove will depend only on the algebraic structure of the group Γ.

The review presents an analysis of results of the Abelian group theory, as well as rings and modules, which concern the definability of algebraic structures by their endomorphism rings and related structures. In the systematization of the results, the greatest attention is paid to torsion-free Abelian groups, which are of particular interest due to the presence of non-isomorphic direct decompositions in this class. This significantly expands the understanding of general, including modern, trends of the development of algebra in the context related to the Baer–Kaplansky theorem. The reflection of the properties of algebraic objects of a certain class in their endomorphism rings is a natural structural connection, the study of which is a separate investigation direction. A striking introduction to this topic was the Baer–Kaplansky theorem for torsion Abelian groups, which dates back to the middle of the last century and states that any isomorphism of endomorphism rings of two groups from this class is inevitably induced by some isomorphism of the groups themselves. Of course, it follows that if two torsion Abelian groups have isomorphic endomorphism rings, then they are isomorphic. This profound result inspired mathematicians to obtain results in the same form concerning other classes of objects. But even in the theory of Abelian groups itself, other classes were discovered for which the analogue of the Baer–Kaplansky theorem is valid. Despite the fundamental difference between the definitions of completely decomposable Abelian groups, which are direct sums of rank-one torsion-free groups, and torsion Abelian groups considered, which are direct sums of finite order cyclic groups, there is one very important common characteristic of these classes: their decompositions into indecomposable summands are determined uniquely up to isomorphism. This property is not possessed by torsion-free Abelian groups in general, whose definability by their endomorphism rings is in the focus of our attention.

Recent advances in minimal topological groups