Finite group extensions of irrational rotations

The ergodicity of certain skew products of irrational rotations of the circle with finite groups is established with application to the construction of “well-distributed sequence generators” for finite groups.     

The quasivariety generated by a torsion-free Abelian-by-finite group

Let L_q(qG) be the quasivariety lattice contained in a quasivariety generated by a group G. It is proved that if G is a finitely generated torsion-free group in (i.e., G is an extension of an Abelian group by a group of exponent 2ⁿ), which is a split extension of an Abelian group by a cyclic group, then the lattice L_q(qG) is a finite chain. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 407–427, July–August, 2007.     

Finite group extensions and the Atiyah conjecture

The Atiyah conjecture for a discrete group states that the -Betti numbers of a finite CW-complex with fundamental group are integers if is torsion-free, and in general that they are rational numbers with denominators determined by the finite subgroups of .

Here we establish conditions under which the Atiyah conjecture for a torsion-free group implies the Atiyah conjecture for every finite extension of . The most important requirement is that is isomorphic to the cohomology of the -adic completion of for every prime number . An additional assumption is necessary e.g. that the quotients of the lower central series or of the derived series are torsion-free.

We prove that these conditions are fulfilled for a certain class of groups, which contains in particular Artin's pure braid groups (and more generally fundamental groups of fiber-type arrangements), free groups, fundamental groups of orientable compact surfaces, certain knot and link groups, certain primitive one-relator groups, and products of these. Therefore every finite, in fact every elementary amenable extension of these groups satisfies the Atiyah conjecture, provided the group does.

As a consequence, if such an extension is torsion-free, then the group ring contains no non-trivial zero divisors, i.e. fulfills the zero-divisor conjecture.

In the course of the proof we prove that if these extensions are torsion-free, then they have plenty of non-trivial torsion-free quotients which are virtually nilpotent. All of this applies in particular to Artin's full braid group, therefore answering question B6 on http://www.grouptheory.info.

Our methods also apply to the Baum-Connes conjecture. This is discussed by Thomas Schick in his preprint ``Finite group extensions and the Baum-Connes conjecture', where for example the Baum-Connes conjecture is proved for the full braid groups.

     

The universal torsion-free image of a group

We show that the universal torsion free homomorphic image of any group given by a sufficiently ‘small’ presentation is locally indicable, and give an application to a conjecture of Levin about equations over torsion free groups. Supported by SERC Visiting Fellowship GR/F 97355.     

Equations in the Q-completion of a torsion-free hyperbolic group

In this paper we prove the algorithmic solvability of finite systems of equations over the Q-completion of a torsion-free hyperbolic group.

     

A group extensions approach to affine relative difference sets of even order

It is shown that a group extensions approach to central relative (k+1,k-1,k,1)-difference sets of even order leads naturally to the notion of an “affine” planar map; a notion analogous to the well-known planar map corresponding to a splitting relative (m,m,m,1)-difference set. Basic properties of affine planar maps are derived and applied to give some new results regarding abelian relative (k+1,k-1,k,1)-difference sets of even order and to give new proofs, in the even order case, for some known results. The paper concludes with computational non-existence results for 10,000<k?100,000.     

Distinguished extensions of a lattice-ordered group

Any lattice-ordered group (l-group for short) is essentially extended by its lexicographic product with a totally ordered group. That is, anl-homomorphism (i.e., a group and lattice homomorphism) on the extension which is injective on thel-group must be injective on the extension as well. Thus nol-group has a maximal essential extension in the categoryIGp ofl-groups withl-homomorphisms. However, anl-group is a distributive lattice, and so has a maximal essential extension in the categoryD of distributive lattices with lattice homomorphisms. Adistinguished extension of onel-group by another is one which is essential inD. We characterize such extensions, and show that everyl-groupG has a maximal distinguished extensionE(G) which is unique up to anl-isomorphism overG.E(G) contains most other known completions in whichG is order dense, and has mostl-group completeness properties as a result. Finally, we show that ifG is projectable then E(G) is the -completion of the projectable hull ofG.Presented by M. Henriksen.     

An example of a torsion-free nilpotent group having no outer automorphisms

A negative answer is given to a question posed by E. Schenkman and F. Haimo: does every torsion-free nilpotent group have an outer automorphism?     

The Jacobson radical of the endomorphism ring of a torsion-free abelian group

We study the action of elements of the Jacobson radical of the endomorphism ring of a torsion-free abelian group on the group itself.