Finitely generated metabelian groups arising from integer polynomials

This paper investigates self-small abelian groups of finite torsion-free rank. We obtain a new characterization of infinite self-small groups. In addition, self-small groups of torsion-free rank 1 and their finite direct sums are discussed.

Soluble Groups of Type (FP)∞ have Finite Torsion-Free Rank

We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval I I . The monic integer transfinite diameter t M ( I ) t_{\mathrm {M}}(I) is defined as the infimum of all such supremums. We show that if I I has length 1 1 , then t M ( I ) = 1 2 t_{\mathrm {M}}(I) = \tfrac {1}{2} . We make three general conjectures relating to the value of t M ( I ) t_{\mathrm {M}}(I) for intervals I I of length less than 4 4 . We also conjecture a value for t M ( [ 0 , b ] ) t_{\mathrm {M}}([0,b]) where 0 > b ≤ 1 0>b\le 1 . We give some partial results, as well as computational evidence, to support these conjectures. We define functions L − ( t ) L_{-}(t) and L + ( t ) L_{+}(t) , which measure properties of the lengths of intervals I I with t M ( I ) t_{\mathrm {M}}(I) on either side of t t . Upper and lower bounds are given for these functions. We also consider the problem of determining t M ( I ) t_{\mathrm {M}}(I) when I I is a Farey interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning this value is true for an infinite family of Farey intervals.

We study when some graded rings are Jacobson rings. In particular we obtain that rings strongly graded either by a polycyclic-by-finite group, or by an abelian group of finite torsion free rank are Jacobson rings in case the identity component is a left Noetherian Jacobson ring. For monoid rings R[S] of abelian monoids of finite rank the same result holds without R being left Noetherian.

Orderable solvable groups in which every relatively convex subgroup is normal are studied. If such a class is subgroup closed than it is precisely the class of solvable orderable groups which are locally of finite (Mal’tsev) rank. A criterion for an orderable metabelian group to have every relatively convex subgroup normal is given. Examples of an orderable solvable group G of length three with periodic G/G′ and of an orderable solvable group of length four with only one proper normal relatively convex subgroup are constructed.

Finite-Rank Free Metabelian Subgroups of Finitely Presented Metabelian Groups

We investigate algebraic sets over certain finitely generated torsion-free metabelian groups. The class of groups under consideration is the class of so-called ρ-groups. It consists of all wreath products of finitely generated free abelian groups and their subgroups. In particular, it includes all free metabelian groups of finite rank. Our main result is a characterization of certain irreducible algebraic sets over ρ-groups. More precisely, we consider irreducible algebraic sets which are determined by a system of equations in n indeterminates. For their coordinate groups, we introduce a discrete invariant called the relative characteristic. This is an ordered pair of non-negative integers. We determine the structure of the coordinate group of the n-dimensional affine space, and show that its relative characteristic is (n, n). Then we characterize the irreducible algebraic sets of relative characteristic (n, n) and (0, k ) where 0 ≤ k ≤ n . We also obtain some examples of somewhat unusual algebraic sets over ρ-groups, thus demonstrating that algebraic sets over these groups are much more varied and complicated than, say, algebraic sets over free groups.

In this paper we show that membership in finitely generated submonoids is undecidable for the free metabelian group of rank 2 and for the wreath product ℤ≀(ℤ×ℤ). We also show that subsemimodule membership is undecidable for finite rank free (ℤ×ℤ)-modules. The proof involves an encoding of Turing machines via tilings. We also show that rational subset membership is undecidable for two-dimensional lamplighter groups.

In this final chapter we give some scattered results on superstable theories. One of the main lines here is the development of some technology which helps towards generalizing results from the finite rank context to the general superstable context. A key point is the interaction between the local theory developed in Chapter 7 and Shelah’s ‘infinity rank’, R∞. This will yield in some situations ‘continuity and definability’ of p-weight. In any case one of the goals of this chapter is to suitably generalize Buechler’s dichotomy theorem to regular types in superstable theories. This is done in section 5, under the assumption that T is countable and has NDOP and NOTOP (two structure properties); this result has great importance for some fine questions concerning the spectrum function. Note we also give some other versions of the dichotomy theorem, which do not depend on assuming NDOP and NOTOP, and which are relevant to Vaught’s conjecture for superstable theories. In the process of developing the technology, we mention various related ‘non-geometric’ results, such as the existence of regular types between arbitrary models, and definability questions in non-multidimensional theories.

Algorithms for linear groups of finite rank

It is proved that a suitable free Abelian group of finite rank is not absolutely closed in the class $ {\mathcal A} $ 2 of metabelian groups. A condition is specified under which a torsion-free Abelian group is not absolutely closed in $ {\mathcal A} $ 2. Also we gain insight into the question when the dominion in $ {\mathcal A} $ 2 of the additive group of rational numbers coincides with this subgroup.

It is proved that an endomorphism of a free metabelian group S r of any finite rank r that preserves primitivity of systems of elements is an automorphism; in addition, every primitive endomorphism of a free solvable group of rank 2 is an automorphism.