Free Metabelian Group Of Rank Research Articles

Orders on free metabelian groups

Abstract A bi-order on a group 𝐺 is a total, bi-multiplication invariant order. A subset 𝑆 in an ordered group ( G , ⩽ ) (G,\leqslant) is convex if, for all f ⩽ g f\leqslant g in 𝑆, every element h ∈ G h\in G satisfying f ⩽ h ⩽ g f\leqslant h\leqslant g belongs to 𝑆. In this paper, we show that the derived subgroup of the free metabelian group of rank 2 is convex with respect to any bi-order. Moreover, we study the convex hull of the derived subgroup of a free metabelian group of higher rank. As an application, we prove that the space of bi-orders of a non-abelian free metabelian group of finite rank is homeomorphic to the Cantor set. In addition, we show that no bi-order for these groups can be recognised by a regular language.

Primitive elements and automorphisms of the free metabelian group of rank 3

Primitive elements and automorphisms of the free metabelian group of rank 3

On the generic triangle group and the free metabelian group of rank 2

We introduce the concept of a generic Euclidean triangle $\tau$ and study the group $G_\tau$ generated by the reflection across the edges of $\tau$. In particular, we prove that the subgroup $T_\tau$ of all translations in $G_\tau$ is free abelian of infinite rank, while the index 2 subgroup $H_\tau$ of all orientation preserving transformations in $G_\tau$ is free metabelian of rank 2, with $T_\tau$ as the commutator subgroup. As a consequence, the group $G_\tau$ cannot be finitely presented and we provide explicit minimal infinite presentations of both $H_\tau$ and $G_\tau$. This answers in the affirmative the problem of the existence of a minimal presentation for the free metabelian group of rank 2. Moreover, we discuss some examples of non-trivial relations in $T_\tau$ holding for given non-generic triangles $\tau$.

On palindromic width of certain extensions and quotients of free nilpotent groups

In [Bardakov and Gongopadhyay, Palindromic width of free nilpotent groups, J. Algebra 402 (2014) 379–391] the authors provided a bound for the palindromic widths of free abelian-by-nilpotent group ANn of rank n and free nilpotent group N n,r of rank n and step r. In the present paper, we study palindromic widths of groups [Formula: see text] and [Formula: see text]. We denote by [Formula: see text] the quotient of the group Gn = 〈x1, …, xn〉, which is free in some variety by the normal subgroup generated by [Formula: see text]. We prove that the palindromic width of the quotient [Formula: see text] is finite and bounded by 3n. We also prove that the palindromic width of the quotient [Formula: see text] is precisely 2(n - 1). As a corollary to this result, we improve the lower bound of the palindromic width of N n,r. We also improve the bound of the palindromic width of a free metabelian group. We prove that the palindromic width of a free metabelian group of rank n is at most 4n - 1.

Systems of elements preserving measure on varieties of groups

It is proved that for any , , a system of elements of a free metabelian group of rank is primitive if and only if it preserves measure on the variety of metabelian groups . From this we obtain the result that a system of elements is primitive in the group if and only if it is primitive in its profinite completion . Furthermore, it is proved that there exist a variety and a nonprimitive element such that preserves measure on .Bibliography: 13 titles.

Tilings and Submonoids of Metabelian 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.

Algebraic sets in 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.

Irreducible Algebraic Sets in Metabelian Groups

We present the construction for a u-product G1 ○ G2 of two u-groups G1 and G2, and prove that G1 ○ G2 is also a u-group and that every u-group, which contains G1 and G2 as subgroups and is generated by these, is a homomorphic image of G1 ○ G2. It is stated that if G is a u-group then the coordinate group of an affine space Gn is equal to G ○ Fn, where Fn is a free metabelian group of rank n. Irreducible algebraic sets in G are treated for the case where G is a free metabelian group or wreath product of two free Abelian groups of finite ranks.

An Automorphism of a Free Metabelian Group Without Fixed Points#

We construct an example of an IA-automorphism of the free metabelian group of rank n ≥ 3 without nontrivial fixed points. That gives a negative answer to the question raised by Shpilrain in [Shpilrain, V. (1998). Fixed points of endomorphisms of a free metabelian group. Math. Proc. Cambridge Philos. Soc. 123(1): 75–83. MR 98k:20056]. By a result of Bachmuth [Bachmuth, S. (1965). Automorphisms of free metabelian groups. Trans. Am. Math. Soc. 118:93–1104. MR 31 #4831], such an automorphism does not exist if the rank is equal to 2.

AUTOMORPHISMS OF PRIME ORDER OF FREE METABELIAN GROUPS

ABSTRACT Let be a prime integer and let be a free metabelian group of rank In the paper an automorphism of order permuting cyclically fixed bases of is studied. The subgroup of fixed points of and the properties of action of on the commutator subgroup of are described. There are also given some results concerning such automorphisms of free abelian groups of rank

Primitive Elements in the Free Metabelian Group of Rank 3

Primitive Elements in the Free Metabelian Group of Rank 3

Fixed points of endomorphisms of a free metabelian group

We consider IA-endomorphisms (i.e. Identical in Abelianization) of a free metabelian group of nite rank, and give a matrix characterization of their xed points which is similar to (yet different from) the well-known characterization of eigenvectors of a linear operator in a vector space. We then use our matrix characterization to elaborate several properties of the xed point groups of metabelian endomorphisms. In particular, we show that the rank of the xed point group of an IA-endomorphism of the free metabelian group of rank n> 2 can be either equal to 0, 1, or greater than (n 1) (in particular, it can be innite). We also point out a connection between these properties of metabelian IA-endomorphisms and some properties of the Gassner representation of pure braid groups.

Automorphic and endomorphic reducibility and primitive endomorphisms of free metabelian groups

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.

Primitivity in the free groups of the variety UmUn

For each m,n >= 0, let Gm,n denote the free group of rank r in the variety UmUn. The main results in this paper are: (i) a necessary and sufficient condition for a system of r elements {v1,…, vr} to form a basis of Gm,ni (ii) necessary and sufficient conditions for a system of l elements {v1,…, vl}, l <= r, to be included in a basis of Gm,0. In particular, (i), (ii) yield corresponding results for the free metabelian group of rank r.

The swap conjecture of tennant and turner

We argue against the conjecture which says that any two finite generating sets for G of the same cardinality are swap equivalent. The latter means that one is changed to another by a finite sequence of generating sets such that all the neighboring sets differ only in a single entry. Namely, it is proved that a free metabelian group of rank 3 has non swap equivalent bases.

Presentations of the Free Metabelian Group of Rank 2

AbstractLet F3 denote the free group of rank 3 and M2 denote the free metabelian group of rank 2. We say that x * F3 is a primitive element of F3 if it can be included a in some basis of F3. We establish the existence of presentations such that N does not contain any primitive elements of F3.

The Mal'tsev-Nielsen equation in a free metabelian group of rank two

Let F 2 denote the free group of rank two with free generators a and b. Nielsen [i, 2] and Mal'tsev [3] have proved'that elements x and y of F 2 are its free generators if and only if the disjunction of equations with respect to z ~ = ~ l z [ x , y ] ~ l = [a, b] has a s o l u t i o n , where [u , v] d e n o t e s t h e c o m m u t a t o r o f e l e m e n t s u and v , i . e . , [u , v] = UVU-Iv -1 The aim of the present note is to prove the analogous result for a free metabelian group of rank two. Let F 2 be a free group of rank two with free generators x I and x~ and F~ 2) be its second commutant, i.e., F~ 2) = [F! I), F~I)], where F! l) = [F2, F2]. Then F2/F! ~) is a free metabelian group of rank two with free generators~ ,.x I and x 2 (we do not distinguish between elements xl, x 2 of F 2 and their images in F2/F[2)). In the proofs we will use derivations of the group ring Z [F~/F~ ~] with values in Z [F/F~I']. All the necessary definitions and properties of these derivations are contained in [4, 5] and detailed proofs can be found in [6-8] and [4]. LEMMA i. If elements u and v of the group)F2/F~ 2) satisfy the equation [u, v] = [xl, x2] , t-~en u and v are free generators of F2/F! 2 . Proof. Direct computations in Z [F21 give the following equations: O[xl, x2]/Oxx = t [ x , z2]x2, 0 [z~, x~l/Ox~ = x~-[z~ , x~]. I f w(xz , x 2) i s an e l e m e n t o f F2 and u and r a r e a r b i t r a r y e l e m e n t s o f F2, t h e n t h e f o l l owing r u l e ( t h e c h a i n r u l e ) h o l d s :

The Free Metabelian Group of Rank Two Contains Continuously Many Nonisomorphic Subgroups

The Free Metabelian Group of Rank Two Contains Continuously Many Nonisomorphic Subgroups

The free metabelian group of rank two contains continuously many nonisomorphic subgroups

It is proved that the free metabelian group of rank two contains continuously many nonisomorphic subgroups.

The Nonfinite Generation of Aut(G), G Free metabelian of Rank 3

The group of automorphisms of the free metabelian group of rank 3 is not finitely generated. Let H be a free solvable group of rank n > 2, Aut(H) the automorphism group of H, and Inn(H) the group of inner automorphisms of H. For n = 2, it was shown by the authors and E. Formanek [4, Theorem 1] that Aut(H)/Inn(H)GL2(Z), Z the ring of integers. One consequence of this fact is that Aut(H) is finitely generated (f.g.). The purpose of this paper is to prove the following contrasting theorem. THEOREM. If G is the free metabelian group of rank three, then Aut(G) is not finitely generated. This result may also be compared with a theorem of L. Auslander [1], which states that the automorphism group of a polycyclic group is finitely presented. If G' denotes the commutator subgroup of G, then the kernel of the natural homomorphism Aut(G) -4 Aut(G/G') is called the group of IA-automorphisms of G and will be denoted by IA(G). IA(G), which contains Inn(G), was shown by the authors [3] to be non-f.g. for G as in the Theorem. To prove that Aut(G) is not f.g., we need to show that IA(G) is not f.g. as an Aut(G/G') -GL3(Z)-operator group. The question arises concerning Aut(H) where H is free metabelian of rank n > 3. The immediate feeling that Aut(H) is also non-f.g. may be incorrect. For definiteness let n = 4. Suppose G is free metabelian of rank 3, and consider Aut(G) as embedded in Aut(H) in an obvious manner. The nonfinite generatiion of Aut(G) comes from the existence of automorphisms in Aut(G), i.e., automorphisms of G F/F which are not induced by automorphisms of the free group F of rank three, and the necessity to include infinitely many nontame automorphisms in any generating set for Aut(G). However, the authors have discovered that many of the nontame automorphisms become tame when considered as elements in Aut(H). Whether this phenomenon is true for all nontame elements of Aut(G) is unknown as yet, but the question of the finite or nonfinite generation of Aut(H) must be considered a difficult open problem, with the interesting possibility of finite generation as the answer. Received by the editors December 31, 1980. 1980 Mathematics Subject Classification. Primary 20F28.