Applications of a New K-Theoretic Theorem to Soluble Group Rings

Kropholler, Peter H.; Linnell, Peter A.; Moody, John Atwell

doi:10.2307/2046771

Cited by 77 publications

(68 citation statements)

References 10 publications

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“…We look over the splitting construction given above. The groups G and H are represented as factor groups of a free group F with basis Note that the majority of rigid solvable groups under consideration are torsion free, and so their integer group rings satisfy a (right) Ore condition [7,8] and are embedded in (right) division rings of quotients [9]. …”

Section: Auxiliary Definitions and Factsmentioning

confidence: 99%

Coproducts of rigid groups

Romanovskii

2011

Algebra Logic

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Let ε = (ε 1 , . . . , ε m ) be a tuple consisting of zeros and ones. Suppose that a group G has a normal series of the formfor ε i = 0, and all factors G i /G i+1 of the series are Abelian and are torsion free as right Z[G/G i ]-modules. Such a series, if it exists, is defined by the group G and by the tuple ε uniquely. We call G with the specified series a rigid m-graded group with grading ε. In a free solvable group of derived length m, the above-formulated condition is satisfied by a series of derived subgroups. We define the concept of a morphism of rigid m-graded groups. It is proved that the category of rigid m-graded groups contains coproducts, and we show how to construct a coproduct G•H of two given rigid m-graded groups. Also it is stated that if G is a rigid m-graded group with grading (1, 1, . . . , 1), and F is a free solvable group of derived length m with basis {x 1 , . . . , x n }, then G • F is the coordinate group of an affine space G n in variables x 1 , . . . , x n and this space is irreducible in the Zariski topology.

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Section: Auxiliary Definitions and Factsmentioning

confidence: 99%

Coproducts of rigid groups

Romanovskii

2011

Algebra Logic

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“…Then ZG is also a left Ore domain, for which there exists a right (and left) division ring of quotients, denoted hereinafter by Q(G). From [5,6], it follows that an integer group ring of a torsion-free soluble group G is a right Ore domain; hence it is embedded in Q(G).…”

Section: 2mentioning

confidence: 99%

Divisible rigid groups

Romanovskii

2008

Algebra Logic

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The concept of a rigid group appeared in studying algebraic geometry over groups that are close to free soluble. In the class of all rigid groups, we distinguish divisible groups the elements of whose quotients G i /G i+1 are divisible by any elements of respective groups rings Z[G/G i ]. It is reasonable to suppose that algebraic geometry over divisible rigid groups is rather well structured. Abstract properties of such groups are investigated. It is proved that in every divisible rigid group H that contains G as a subgroup, there is a minimal divisible subgroup including G, which we call a divisible closure of G in H. Among divisible closures of G are divisible completions of G that are distinguished by some natural condition. It is shown that a divisible completion is defined uniquely up to G-isomorphism.

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“…Then ZG is also a left Ore domain, for which there exists a right (and left) division ring of quotients, which we denote by Q(G). It follows from [13,14] that an integer group ring of a torsion-free soluble group G is a right Ore domain; hence it is embedded in Q(G).…”

Section: 2mentioning

confidence: 99%