Groups elementarily equivalent to a finitely generated free metabelian group

We study the augmentation quotients of the IA-automorphism group of a free group and a free metabelian group. First, for any group G , we construct a lift of the k ‐th Johnson homomorphism of the automorphism group of G to the k ‐th augmentation quotient of the IA-automorphism group of G . Then we study the images of these homomorphisms for the case where G is a free group and a free metabelian group. As a corollary, we detect a Z‐free part in each of the augmentation quotients, which can not be detected by the abelianization of the IA-automorphism group. 20F28; 16S34

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.

Endomorphisms preserving an orbit in a relatively free metabelian group

Let S be the free metabelian group of rank 2. In this paper we prove the following results:(i) Given a pair of elements g, h of S, there exists an algorithm to decide whether or not g is an automorphic image of h; (ii) If g, h are in the commutator subgroup S′ of S such that each is an endomorphic image of the other then g , h are automorphic; (iii) If an endomorphism of S maps primitive elements of S to primitive elements of S then it defines an automorphism of S. We also include an example to show that, in (ii) above,the requirement that g,h are in S ? can not be relaxed.

Finite-Rank Free Metabelian Subgroups of Finitely Presented Metabelian Groups

The research launched in [1] is brought to a close by examining algebraic sets in a metabelian group G in two important cases: (1) G = Fn is a free metabelian group of rank n; (2) G = Wn,k is a wreath product of free Abelian groups of ranks n and k.

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.

We study the free metabelian group $M(2,n)$ of prime power exponent n on two generators by means of invariants $M(2,n)'\to \mathbb {Z}_n$ that we construct from colorings of the squares in the integer grid $\mathbb {R} \times \mathbb {Z} \cup \mathbb {Z} \times \mathbb {R}$ . In particular, we improve bounds found by Newman for the order of $M(2,2^k)$ . We study identities in $M(2,n)$ , which give information about identities in the Burnside group $B(2,n)$ and the restricted Burnside group $R(2,n)$ .

An embedding theorem for finitely generated free metabelian groups

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.

An embedding theorem for finitely generated free metabelian groups

GLn-1(R), the group of invertible (n - 1) × (n - 1) matrices with entries in R = ℤ [x1, …,xm] or K[x1, …,xm+1], m ≥ 1, the polynomial ring over the integers ≥ or a field K.

Introduction. For n_2 and p a prime, let Mn,,2 denote the free metabelian group of exponent p2 on n generators. The precise nilpotency class of M.,4 (n> 2) was established in Gupta and Tobin [2 ], where it was also shown that I M2,41 = 2 10 and M33,41 = 23. In a recent paper [1], Bachmuth and Mochizuki have shown that the class of M2,,2 (p odd) is precisely 2(p2-p). In another paper (to appear) Bachmuth, Heilbronn and Mochizuki have shown that for 3<k<p+1, the class of iIk,p2 is 2 (p2-p) or 2(p2-p)+1. In this paper we complete the discussion for the case k _ p +2.

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.

This paper explores various connections between combinatorial group theory, semigroup theory, and formal language theory. Let G = &lt;A|R&gt; be a group presentation and ℬA, R its standard 2-complex. Suppose X is a 2-complex with a morphism to ℬA, R which restricts to an immersion on the 1-skeleton. Then we associate an inverse monoid to X which algebraically encodes topological properties of the morphism. Applications are given to separability properties of groups. We also associate an inverse monoid M(A, R) to the presentation &lt;A|R&gt; with the property that pointed subgraphs of covers of ℬA, R are classified by closed inverse submonoids of M(A, R). In particular, we obtain an inverse monoid theoretic condition for a subgroup to be quasiconvex allowing semigroup theoretic variants on the usual proofs that the intersection of such subgroups is quasiconvex and that such subgroups are finitely generated. Generalizations are given to non-geodesic combings. We also obtain a formal language theoretic equivalence to quasiconvexity which holds even for groups which are not hyperbolic. Finally, we illustrate some applications of separability properties of relatively free groups to finite semigroup theory. In particular, we can deduce the decidability of various semidirect and Mal/cev products of pseudovarieties of monoids with equational pseudovarieties of nilpotent groups and with the pseudovariety of metabelian groups.