Wreath Product Construction Research Articles

On the embedding of Galois groups into wreath products

In this paper we make explicit an application of the wreath product construction to the Galois groups of field extensions. More precisely, given a tower of fields F ⊆ K ⊆ L with L/F finite and separable, we explicitly construct an embedding of the Galois group Gal ( L c / F ) into the regular wreath product Gal ( L c / K c ) ≀ r Gal ( K c / F ) . Here Lc (resp. Kc ) denotes the Galois closure of L/F (resp. K/F). Similarly, we also construct an explicit embedding of the Galois group Gal ( L c / F ) into the smaller sized wreath product Gal ( L c / K ) ≀ Ω Gal ( K c / F ) , where Ω = Hom F ( K , K c ) is acted on by composition of automorphisms in Gal ( K c / F ) . Moreover, when L/K is a Kummer extension we prove a sharper embedding, that is, that Gal ( L c / F ) embeds into the wreath product Gal ( L / K ) ≀ Ω Gal ( K c / F ) . As corollaries we obtain embedding theorems when L/K is cyclic and when it is quadratic with char ( F ) ≠ 2 . We also provide examples of these embeddings and as an illustration of the usefulness of these embedding theorems, we survey some recent applications of these types of results in field theory, arithmetic statistics, number theory and arithmetic geometry.

Superrigidity for dense subgroups of lie groups and their actions on homogeneous spaces

An essentially free group action \(\Gamma \curvearrowright (X,\mu )\) is called W\(^*\)-superrigid if the crossed product von Neumann algebra \(L^\infty (X) \rtimes \Gamma \) completely remembers the group \(\Gamma \) and its action on \((X,\mu )\). We prove W\(^*\)-superrigidity for a class of infinite measure preserving actions, in particular for natural dense subgroups of isometries of the hyperbolic plane. The main tool is a new cocycle superrigidity theorem for dense subgroups of Lie groups acting by translation. We also provide numerous countable type II\(_1\) equivalence relations that cannot be implemented by an essentially free action of a group, both of geometric nature and through a wreath product construction.

Universal covers of finite groups

Motivated by the success of quotient algorithms, such as the well-known p-quotient or solvable quotient algorithms, in computing information about finite groups, we describe how to compute finite extensions H˜ of a finite group H by a direct sum of isomorphic simple ZpH-modules such that H and H˜ have the same number of generators. Similar to other quotient algorithms, our description will be via a suitable covering group of H. Defining this covering group requires a study of the representation module, as introduced by Gaschütz in 1954. Our investigation involves so-called Fox derivatives (coming from free differential calculus) and, as a by-product, we prove that these can be naturally described via a wreath product construction. An important application of our results is that they can be used to compute, for a given epimorphism G→H and simple ZpH-module V, the largest quotient of G that maps onto H with kernel isomorphic to a direct sum of copies of V. For this we also provide a description of how to compute second cohomology groups for the (not necessarily solvable) group H, assuming a confluent rewriting system for H. To represent the corresponding group extensions on a computer, we introduce a new hybrid format that combines this rewriting system with the polycyclic presentation of the module.

LOCALLY COMPACT WREATH PRODUCTS

Wreath products of nondiscrete locally compact groups are usually not locally compact groups, nor even topological groups. As a substitute introduce a natural extension of the wreath product construction to the setting of locally compact groups. Applying this construction, we disprove a conjecture of Trofimov, constructing compactly generated locally compact groups of intermediate growth without any open compact normal subgroup.

Algebras and semigroups of locally subexponential growth

We prove that a countable dimensional associative algebra (resp. a countable semigroup) of locally subexponential growth is M∞-embeddable as a left ideal in a finitely generated algebra (resp. semigroup) of subexponential growth. Moreover, we provide bounds for the growth of the finitely generated algebra (resp. semigroup). The proof is based on a new construction of matrix wreath product of algebras.

EMBEDDABILITY OF GENERALISED WREATH PRODUCTS

AbstractGiven two finitely generated groups that coarsely embed into a Hilbert space, it is known that their wreath product also embeds coarsely into a Hilbert space. We introduce a wreath product construction for general metric spaces $X,Y,Z$ and derive a condition, called the (${\it\delta}$-polynomial) path lifting property, such that coarse embeddability of $X,Y$ and $Z$ implies coarse embeddability of $X\wr _{Z}Y$. We also give bounds on the compression of $X\wr _{Z}Y$ in terms of ${\it\delta}$ and the compressions of $X,Y$ and $Z$.

Wreath products by a leavitt path algebra and affinizations

We introduce ring theoretic constructions that are similar to the construction of wreath product of groups [M. Kargapolov and Y. Merzlyakov, Fundamentals of the Theory of Groups (Springer-Verlag, New York, 1979)]. In particular, for a given graph Γ = (V, E) and an associate algebra A, we construct an algebra B = A wr L(Γ) with the following property: B has an ideal I, which consists of (possibly infinite) matrices over A, B/I ≅ L(Γ), the Leavitt path algebra of the graph Γ. Let W ⊂ V be a hereditary saturated subset of the set of vertices [G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293(2) (2005) 319–334], Γ(W) = (W, E(W, W)) is the restriction of the graph Γ to W, Γ/W is the quotient graph [G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293(2) (2005) 319–334]. Then L(Γ) ≅ L(W) wr L(Γ/W). As an application we use wreath products to construct new examples of (i) affine algebras with non-nil Jacobson radicals, (ii) affine algebras with non-nilpotent locally nilpotent radicals.

Bilinear factorization of algebras

Bilinear factorization of algebras

Wreath Products of Permutation Classes

A permutation class which is closed under pattern involvement may be described in terms of its basis. The wreath product construction $X\wr Y$ of two permutation classes $X$ and $Y$ is also closed, and we exhibit a family of classes $Y$ with the property that, for any finitely based class $X$, the wreath product $X\wr Y$ is also finitely based. Additionally, we indicate a general construction for basis elements in the case where $X\wr Y$ is not finitely based.

Free Wreath Product by the Quantum Permutation Group

Let A be a compact quantum group, let n∈N * and let A aut(X n ) be the quantum permutation group on n letters. A free wreath product construction A*w A aut(X n ) is introduced. This construction provides new examples of quantum groups, and is useful to describe the quantum automorphism group of the n-times disjoint union of a finite connected graph.

Rooted wreath products

We introduce “rooted valuation products” and use them to construct universal Abelian lattice-ordered groups (with prescribed set of components) from the more classical theory of H. Hahn. The wreath product construction of W.C. Holland and S.H. McCleary generalised the Abelian (lattice-ordered) permutation group ideas to give universals for transitive (ℓ-)permutation groups with prescribed set of primitive components. In the case of (not necessarily transitive) sublattice subgroups of order-preserving permutations of totally ordered sets, the set of natural congruences forms a root system. We generalise the rooted valuation product construction to the permutation case when all natural primitive components are regularly obtained; we analogously obtain universals for these permutation groups (for a prescribed set of natural primitive components) which we call “rooted wreath products.” We identify the rooted valuation product with an appropriate subgroup of the corresponding rooted wreath product. The maximal Abelian group actions on the ordered real line were characterised in by R. Winkler, and their digital representations were consequently obtained. We use the rooted wreath product construction to get a more general result, and deduce Winkler's characterisation as a consequence.

Restricted permutations and the wreath product

Restricted permutations are those constrained by having to avoid subsequences ordered in various prescribed ways. A closed set is a set of permutations all satisfying a given basis set of restrictions. A wreath product construction is introduced and it is shown that this construction gives rise to a number of useful techniques for deciding the finite basis question and solving the enumeration problem. Several applications of these techniques are given.

Wreath products along the period-doubling route to chaos

The wreath-product construction is used to give a complete combinatorial description of the dynamics of period-doubling quadratic maps leading to the Feigenbaum map. An explicit description of the action on periodic points uses the Thue–Morse sequence. In particular, a wreath-product construction of this sequence is given. The combinatorial renormalization operator on the period-doubling family of maps is invertible.

Products of idempotent endomorphisms of free acts of infinite rank

In 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (i.e., composite) of idempotent self-maps of that set. Using a wreath product construction introduced by V. Fleischer, the first-named author was recently able to describe products of idempotent endomorphisms of a freeS-act of finite rank whereS is any monoid. The purpose of the present paper is to extend this result to freeS-acts of infinite rank.

Uncountable existentially closed groups in locally finite group classes

In this paper, will always denote a local class of locally finite groups, which is closed with respect to subgroups, homomorphic images, extensions, and with respect to cartesian powers of finite -groups. Examples for x are the classes L ℐπ of all locally finite π-groups and L(ℐπ ∩ ) of all locally soluble π-groups (where π is a fixed set of primes). In [4], a wreath product construction was used in the study of existentially closed -groups (=e.c. -groups); the restrictive type of construction available in [4] permitted results for only countable groups. This drawback was then removed partially in [5] with the help of permutational products. Nevertheless, the techniques essentially only permitted amalgamation of -groups with locally nilpotent π-groups. Thus, satisfactory results could be obtained for Lp-groups (resp. locally nilpotent π-groups) [6], while the theory remained incomplete in all other cases.

The persistence of universal formulae in free algebras

Gilbert Baumslag, B.H. Neumann, Hanna Neumann, and Peter M. Neumann successfully exploited their concept of discrimination to obtain generating groups of product varieties via the wreath product construction. We have discovered this same underlying concept in a somewhat different context. Specifically, let V be a non-trivial variety of algebras. For each cardinal α let Fα(V) be a V-free algebra of rank α. Then for a fixed cardinal r one has the equivalence of the following two statements:(1) Fr(V) discriminates V. (1*) The Fs(V) satisfy the same universal sentences for all s≥r. Moreover, we have introduced the concept of strong discrimination in such a way that for a fixed finite cardinal r the following two statements are equivalent:(2) Fr(V) strongly discriminates V. (2*) The Fs(V) satisfy the same universal formulas for all s ≥ r whenever elements of Fr(V) are substituted for the unquantified variables. On the surface (2) and (2*) appear to be stronger conditions than (1) and (1*). However, we have shown that for particular varieties (of groups) (2) and (2*) are no stronger than (1) and (1*).

On the 3j symmetries

The symmetry properties of the 3jm tensor for any finite or compact linear group are discussed using a wreath product construction. This is shown to provide a complete group theoretic explanation for all symmetry properties whether ‘‘essential’’ or ‘‘arbitrary.’’ The link with the similar—but distinct—method of inner plethysms is considered.

Corrections and Additions to “Some Applications of the Wreath Product Construction”

Corrections and Additions to “Some Applications of the Wreath Product Construction”

Corrections and Additions to "Some Applications of the Wreath Product Construction"

Corrections and Additions to "Some Applications of the Wreath Product Construction"

Some Applications of the Wreath Product Construction

Some Applications of the Wreath Product Construction