On algebraic sets over 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.

It is proved that every consistent system of equations over a free semilattice of arbitrary rank is equivalent to its finite subsystem. Furthermore, irreducible algebraic sets are studied, and we look at the consistency problem for systems of equations over free semilattices.

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

Long colimits of topological groups II: Free groups and vector spaces

We describe groups elementarily equivalent to a free metabelian group with n generators. We also explore an exponentiation that naturally occurs in metabelian groups.

We will construct an almost free non-free group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of cardinality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 1"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> ℵ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\aleph _{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G slash upper G prime"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>G</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">G/G’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a free abelian group.

Using the wonderful compactification of a semisimple adjoint affine algebraic\ngroup G defined over an algebraically closed field k of arbitrary\ncharacteristic, we construct a natural compactification Y of the G-character\nvariety of any finitely generated group F. When F is a free group, we show that\nthis compactification is always simply connected with respect to the \\'etale\nfundamental group, and when k=C it is also topologically simply connected. For\nother groups F, we describe conditions for the compactification of the moduli\nspace to be simply connected and give examples when these conditions are\nsatisfied, including closed surface groups and free abelian groups when\nG=PGL(n,C). Additionally, when F is a free group we identify the boundary\ndivisors of Y in terms of previously studied moduli spaces, and we construct a\nfamily of Poisson structures on Y and its boundary divisors arising from\nBelavin-Drinfeld splittings of the double of the Lie algebra of G. In the\nappendix, authored by Sam Evens and Arlo Caine, we explain how to put a Poisson\nstructure on a quotient of a Poisson algebraic variety by the action of a\nreductive Poisson algebraic group.\n

A Bieberbach group is a torsion free crystallographic group that represents an extension of a free abelian lattice group by a finite point group. This research began by taking the group offered in the Crystallographic Algorithms and Tables (CARAT) package, which is in the matrix form. There are only four Bieberbach groups of dimension six to be isomorphic to the quaternion point group of order eight. In this study, three Bieberbach groups of dimension six with the quaternion point group of order eight that are considered as only the first group has been found its well-defined polycyclic presentation. Every group has eight generators that describe the group. However, the algorithm used in constructing the polycyclic presentation requires a new arbitrary generator to be added into the group. Then the consistency relations need to be checked and the polycyclic presentation is said to be a well-defined construction if it is consistent. Therefore, this study shows the construction of polycyclic presentation with the new arbitrary generator for all three groups. Furthermore, the polycyclic presentation for the second group has been proven to be consistent, which implies that the construction is well-defined.

Finite-Rank Free Metabelian Subgroups of Finitely Presented Metabelian Groups

We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various sub- groups of the mapping class group—the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration—and additionally to the outer automorphism group of a free group and to certain linear groups. There are many analogies and interconnections between the theories of right- angled Artin groups on one hand and mapping class groups on the other hand. For instance, by the work of Crisp and Wiest (13), the work of Koberda (27), and the work of the first two authors with Mangahas (12), there is an abundance of injective homomorphisms from right-angled Artin groups to mapping class groups. Also, the last two authors proved (30) that any two elements of the pure braid group either generate a free group or a free abelian group—a property shared by all right-angled Artin groups (4, Theorem 1.2). We are thus led to ask to what extent mapping class groups are the same as right-angled Artin groups. It is straightforward to see that most mapping class groups are not isomorphic to right-angled Artin groups, for instance because right-angled Artin groups are torsion free. On the other hand, mapping class groups have finite-index sub- groups that are torsion free, and so this leaves open the possibility that mapping class groups are abstractly commensurable to right-angled Artin groups, that is, that they have isomorphic finite-index subgroups. We prove that, aside from a small number of exceptions, this is not the case. We also extend this result to several classes of groups related to mapping class groups. We start by recalling some definitions. To a finite graph , we can associate a right-angled Artin grou p: this is the group with one generator for each vertex of , and one defin ing relator for each edge, namely, the commutator of the two generators corresponding to the endpoints. Let Sg,n denote a closed, connected, orientable surface of genus g with n marked points. The mapping class group Mod(Sg,n) is the group of homotopy classes of orientation-preserving homeomorphisms of Sg,n preserving the set of marked points. As discussed in Koberda's paper (27, Theorem 1.5), no finite-index subgroup of Mod(Sg,n) injects into a right-angled Artin group if g ≥ 2 and (g,n) 6 (2,0); see

The notion of an equation over a profinite group is defined, as well as the concepts of an algebraic set and of a coordinate group. We show how to represent the coordinate group as a projective limit of coordinate groups of finite groups. It is proved that if the set π(G) of prime divisors of the profinite period of a group G is infinite, then such a group is not Noetherian, even with respect to one-variable equations. For the case of Abelian groups, the finiteness of a set π(G) gives rise to equational Noetherianness. The concept of a standard linear pro-p-group is introduced, and we prove that such is always equationally Noetherian. As a consequence, it is stated that free nilpotent pro-p-groups and free metabelian pro-p-groups are equationally Noetherian. In addition, two examples of equationally non-Noetherian pro-p-groups are constructed. The concepts of a universal formula and of a universal theory over a profinite group are defined. For equationally Noetherian profinite groups, coordinate groups of irreducible algebraic sets are described using the language of universal theories and the notion of discriminability.

On λ-homomorphic skew braces

The group of 2-by-2 matrices with integer entries and determinant $\pm > 1$ can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, namely the general linear groups ${\rm{GL}}(n,\Z)$, the groups of outer automorphisms of free groups of rank $n\geq 2$, and the mapping class groups of closed orientable surfaces of genus $g\geq 1$. Much of the work on mapping class groups and automorphisms of free groups is motivated by the idea that these sequences of groups are strongly analogous. In this article we highlight a few of the most striking similarities and differences between these series of groups and present a list of open problems motivated by this philosophy.

A description of the general solution of given bounded periodicity exponent is obtained for an arbitrary system of equations in a free group. On the basis of this result an algorithm is constructed that computes the rank of coefficient-free systems of equations. Bibliography: 7 titles.

The relationship between the Wiener-Hopf factorisation of matrices and the solution of systems of certain operator equations is discussed in an algebraic setting. It is shown that the study of the waiting time process of the nth arriving group of the general single server bulk queue leads to equations of that type. This system of equations may be considered as an extension of Lindley's waiting-time equation.