Jordan tori for a torsion free abelian group

We classify Jordan G-tori, where G is any torsion-free abelian group. Using the Zelmanov prime structure theorem, such a class divides into three types, the Hermitian type, the Clifford type, and the Albert type. We concretely describe Jordan G-tori of each type.     

The torsion group of a field defined by radicals

Let $\text{L}\text{F}$ be a finite separable extension, $\text{L}{}^1\text{= L{0}}$, and $\text{T(}\text{L}{}^1\text{F}{}^1\text{)}$ the torsion subgroup of $\text{L}{}^1\text{F}{}^1$. When $\text{L}\text{F}$ is an abelian extension $\text{T(}\text{L}{}^1\text{F}{}^1\text{)}$ is explicitly determined. This information is used to study the structure of $\text{T(}\text{L}{}^1\text{F}{}^1\text{)}$. In particular, $\text{T(}\text{F(α)}{}^1\text{F}{}^1\text{)}$ when a^m = a ∈ F is explicitly determined.     

紧群作用下的扭化解析挠率

在紧群中的非单位元的作用只有孤立的不动点的情形下, 我们研究了扭化de Rham 复形上的解析挠率的等变情形. 在一些条件下, 我们也研究了Z₂- 分次的椭圆复形上的离域的(delocalized) L²-解析挠率.     

On the torsion of group extensions and group actions

In this note, we study the torsion of extensions of finitely generated abelian by elementary abelian groups. When the action is trivial , we make a specific choice of a 1-cochain for a vanishing multiple of the cohomology class defining the extension and use it to completely describe the torsion of central extensions. As an application, one gets that, under the assumption of trivial action on homology, Z_p^r may act freely on (S¹)^k if and only if r?k, providing an alternative proof of the main theorem in [Trans. Amer. Math. Soc. 352 (6) (2000) 2689-2700] for central extensions.     

Abelian torsion groups with a countably compact group topology

Comfort and Remus [W.W. Comfort, D. Remus, Abelian torsion groups with a pseudocompact group topology, Forum Math. 6 (3) (1994) 323–337] characterized algebraically the Abelian torsion groups that admit a pseudocompact group topology using the Ulm–Kaplansky invariants.We show, under a condition weaker than the Generalized Continuum Hypothesis, that an Abelian torsion group (of any cardinality) admits a pseudocompact group topology if and only if it admits a countably compact group topology. Dikranjan and Tkachenko [D. Dikranjan, M. Tkachenko, Algebraic structure of small countably compact Abelian groups, Forum Math. 15 (6) (2003) 811–837], and Dikranjan and Shakhmatov [D. Dikranjan, D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on Abelian groups, Topology Appl. 151 (1–3) (2005) 2–54] showed this equivalence for groups of cardinality not greater than .We also show, from the existence of a selective ultrafilter, that there are countably compact groups without non-trivial convergent sequences of cardinality κ^ω, for any infinite cardinal κ. In particular, it is consistent that for every cardinal κ there are countably compact groups without non-trivial convergent sequences whose weight λ has countable cofinality and λ>κ.     

Wiener-Hopf operators on a subsemigroup of a discrete torsion free abelian group

In the paper Wiener-Hopf operators on a semigroup of nonnegative elements of a linearly quasi-ordered torsion free Abelian group are considered. Wiener-Hopf factorization of an invertible element of the group algebra is constructed, notions of a topological index and a factor index are introduced. It turns out that the set of factor indices for invertible elements of the group algebra is a linearly ordered group. It is shown that Wiener-Hopf operator with an invertible symbol is an one-side invertible operator and its invertibility properties are defined by the sign of the factor index of its symbol. Groups on which there exist nontrivial Fredholm Wiener-Hopf operators are described. As an example, all linear quasi-orders on the group ⁿ are found and corresponding Wiener-Hopf operators are considered.     

Some examples of torsion in the Griffiths group

Partially supported by an NSF Postdoctoral Fellowship and by the Max-Planck-Institut für Mathematik     

Reidemeister torsion and group invariants of three-dimensional manifolds

Two formulas are presented in this note. The first is purely algebraic and expresses the first elementary ideal of a finitely generated group in terms of the module of elementary derivatives. The second formula expresses the module of elementary derivatives of the fundamental group of a connected compact three-dimensionalpl -manifold with zero Euler characteristic in terms of the Reidemeister torsion of this manifold.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 66, pp. 204–205, 1976.     

Specialization of the torsion subgroup of the Chow group

An example is given in which specialization is not injective. Partial support by the NSF (DMS-0200012) gratefully acknowledged.     

Hirsch induction and torsion units in group rings

Supported by Canadian NSERC Grant A-5300 and Polish Scientific Grant 211399101.     

The bounded Lie Engel property on torsion group algebras

Let F be a field of characteristic different from 2 and G a group with involution ∗. Extend the involution to the group ring FG, and write (FG)⁻ for the Lie subalgebra of FG consisting of the skew elements. We classify the torsion groups G having no elements of order 2 such that (FG)⁻ is bounded Lie Engel.     

The finite torsion subgroup of an LCA group need not split

A locally compact abelian topological groupG is constructed whose maximal torsion subgroup is finite but not a topological direct summand ofG.     

On the normalizer problem for integral group rings of torsion groups

In this paper, we investigate the normalizer property for the integral group ring of a torsion group. We show that this property holds for locally finite nilpotent groups. A necessary and sufficient condition for this property to hold for any torsion group is also given. This research was supported in part by a research grant from the Natural Sciences and Engineering Research Council of Canada.