Solvable Groups Of Finite Rank Research Articles

The cohomology of virtually torsion-free solvable groups of finite rank

Assume that $G$ is a virtually torsion-free solvable group of finite rank and $A$ a $\mathbb ZG$-module whose underlying abelian group is torsion-free and has finite rank. We stipulate a condition on $A$ that ensures that $H^n(G,A)$ and $H_n(G,A)$ are finite for all $n\geq 0$. Using this property for cohomology in dimension two, we deduce two results concerning the presence of near supplements and complements in solvable groups of finite rank. As an application of our near-supplement theorem, we obtain a new result regarding the homological dimension of solvable groups.

Torsion cohomology for solvable groups of finite rank

We define a class U of solvable groups of finite abelian section rank which includes all such groups that are virtually torsion-free as well as those that are finitely generated. Assume that G is a group in U and A a ZG-module. If A is Z-torsion-free and has finite Z-rank, we stipulate a condition on A that guarantees that Hn(G,A) and Hn(G,A) must be finite for n≥0. Moreover, if the underlying abelian group of A is a Černikov group, we identify a similar condition on A that ensures that Hn(G,A) must be a Černikov group for all n≥0.

‘COHOMOLOGY AND PROFINITE TOPOLOGIES FOR SOLVABLE GROUPS OF FINITE RANK’

AbstractWe remedy an omission in the proof of Proposition 2.7 of the paper ‘Cohomology and profinite topologies for solvable groups of finite rank’, Bull. Aust. Math. Soc.86 (2012), 254–265. This proposition states that a solvable group with finite abelian section rank has merely finitely many subgroups of any given index.

Locally Indicable Groups of Finite Mal’tsev Rank

Recall that G is called a group of finite Mal’tsev rank r if every finitely generated subgroup of G is a group with r generators and r is the least number with this property. Below by the word ‘rank’ we mean the Mal’tsev rank. The notation and terminology adopted here are borrowed from [1, 2]. It is well known that the class of groups of finite rank is closed under subgroups, homomorphic images, and direct products of finitely many groups. For torsion-free Abelian groups, the standard concept of the rank of an Abelian group coincides with that of the Mal’tsev rank. We are also well aware that every torsion-free locally nilpotent group of finite rank is nilpotent and its nilpotency class does not exceed its rank. Groups of finite rank with additional restrictions were studied by many mathematicians such as S. N. Chernikov, V. S. Charin, and D. I. Zaitsev. Research on groups of finite rank was reduced, as a rule, to imposing on a group conditions close to solvability. An impetus for writing this paper is the question whether every linearly orderable finitely generated group will be solvable. The structure of solvable matrix groups over Q is well known. By a theorem of A. I. Mal’tsev, these are finite extensions of triangular matrix groups over Q. We know that every torsion-free solvable group of finite rank can be represented by matrices over the field of rational numbers [1]. Note also that if the rank of a torsion-free solvable matrix group over Q is finite then it is equal to the rational rank (D. I. Zaitsev). In looking into the above question in detail, it became clear that its resolution reduces to treating a similar question for a wider class of groups, namely, the class of locally indicable groups. Recall that a group G is said to be locally indicable if every nontrivial finitely generated subgroup of G admits a homomorphism onto an infinite cyclic group. This class of groups has interesting group-theoretic properties. In [3], it was proved that every one-relator group having no elements of finite order is locally indicable.

Solvable quotients of Kähler groups

AbstractWe prove several results on the structure of solvable quotients of funda-mental groups of compact Ka¨hler manifolds (Ka¨hler groups). 1. Introduction. We first recall a definition from [AN].Definition 1.1 A solvable group Γ has finite rank, if there is a decreasing sequenceΓ = Γ 0 ⊃ Γ 1 ⊃ ... ⊃ Γ m+1 = {1} of subgroups, each normal in its predecessor, suchthat Γ i /Γ i+1 is abelian and Q ⊗(Γ i /Γ i+1 ) is finite dimensional for all i.In what follows F r denotes a free group with the number of generators r ∈ Z + ∪{∞}.Our main result isTheorem 1.2 Let M be a compact Kahler manifold. Assume that the fundamentalgroup π 1 (M) is defined by the sequence{1} −→ F −→ π 1 (M)−→ p H −→ {1}where H is a solvable group of finite rank of the form{0} −→ A −→ H −→ B −→ {0}with non-trivial abelian groups A,B so that Q ⊗ A ∼= Q m and m ≥ 1. Assume alsothat p −1 (A) ⊂ π 1 (M) does not admit a surjective homomorphism onto F ∞ . Then alleigen-characters of the conjugate action of B on the vector space Q⊗A are torsion.In Lemma 2.3 we will show that the condition for p

Condition min-?-N and the representations of solvable groups connected with IT

We study just infinite JG-modules, where J is either the ℤ -group algebra or the F〈t〉-group algebra of the infinite cyclic group 〈t〉 over a finite field F and G is a solvable group of finite rank. With the help of the obtained results it is proved that the finitely approximated torsionfree solvable groups with the condition Min-∞- N are minimax.

Elementary group equivalence with the integral length function

The paper determines criteria of elementary equivalence for some classes of free groups with operators and free products with the length function. The case of a group with operators admitting rational coordinatization with a finite basis is completely analyzed. They are polycyclic, solvable groups of finite rank without torsion, and Chernikov groups. The concept of $\omega$-isomorphism of groups intermediate between elementary equivalence and isomorphism is important for the aspects of elementary equivalence of groups with operators and free product. it is proved that $\omega$-isomorphism of arbitrary groups of operators is followed by the elementary equivalence of the respective free operator groups (free products) with length function.

A Finite Index Property of Certain Solvable Groups

AbstractA group G is said to have the FINITE INDEX property (G is an FI-group) if, whenever H≤G, xp ∈ H for some x in G and p > 0, then |〈H, x〉: H| is finite. Following a brief discussion of some locally nilpotent groups with this property, it is shown that torsion-free solvable groups of finite rank which have the isolator property are FI-groups. It is deduced from this that a finitely generated torsion-free solvable group has an FI-subgroup of finite index if and only if it has finite rank.

The ranks of the p-central series of free groups of certain manifolds

A factor basis is found for the p-central series of a free torsion-free group for which a factor basis of the lower central series can be chosen among the basis commutators. For free solvable groups of finite rank, recursion formulas are obtained for the calculation of the ranks of the factors of the p-central series.

Solvable groups of finite rank

Solvable groups of finite rank

Locally solvable groups of finite rank. II

Locally solvable groups of finite rank. II

ON A PROBLEM OF MAL'CEV

The problem of A. I. Mal'cev on the structure of linear groups of finite rank is solved: a linear group over a field is a group of finite rank if and only if it is a finite extension of a solvable group of finite rank. Bibliography: 8 titles.