Wreath Product Research Articles

On distinguishing labelling of sets under the wreath product action

On distinguishing labelling of sets under the wreath product action

A note on the Kaloujnine-Krasner theorem

The celebrated Kaloujnine-Krasner theorem associates, with a short exact sequence of groups and a section , an embedding of G into the (unrestricted) wreath product of N and H. Given two groups H and N, a short exact sequence as above is called an extension of H by N, denoted by . Moreover, one says that two extensions and of H by N are equivalent if there exists a group isomorphism such that and . We say that two embeddings and are equivalent if there exists a group isomorphism such that . We show that two extensions and are equivalent if and only if the embeddings and , associated with any two sections and via the Kaloujnine-Krasner theorem, are equivalent.

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.

Property FW and wreath products of groups: A simple approach using Schreier graphs

The group property FW stands in-between the celebrated Kazhdan’s property (T) and Serre’s property FA. Among many characterizations, it might be defined, for finitely generated groups, as having all Schreier graphs one-ended.It follows from the work of Y. Cornulier that a finitely generated wreath product G≀XH has property FW if and only if both G and H have property FW and X is finite. The aim of this paper is to give an elementary, direct and explicit proof of this fact using Schreier graphs.

Note on bi-exactness for creation operators on Fock spaces

In this note, we introduce and study a notion of bi-exactness for creation operators acting on full, symmetric and anti-symmetric Fock spaces. This is a generalization of our previous work, in which we studied the case of anti-symmetric Fock spaces. As a result, we obtain new examples of solid actions as well as new proofs for some known solid actions. We also study free wreath product groups in the same context.

Constructing group inclusions with arbitrary depth via wreath products

It was proved in a paper by Burciu, Kadison and Külshammer in 2011 that the ordinary depth d(S_n,S_{n+1}) of the symmetric group S_n in S_{n+1} is 2n-1, so arbitrarily large odd numbers can occur as subgroup depth. Lars Kadison in 2011 posed the question if subgroups of even ordinary depth bigger than 6 can occur. Recently in a paper with Breuer we constructed a series (G_n,H_n) of groups and subgroups where the depth d(H_n,G_n) was 2n, thus answering the question of Kadison. Here we generalize the method of that proof. The main result of this paper is that for every positive integer n there are infinitely many pairs (G, H) of finite groups such that d(H,G)=n. As a corollary of its proof we get that for every positive integer n there are infinitely many triples (H, N, G) of finite solvable groups Htriangleleft Ntriangleleft G such that G/N is cyclic of order lceil {n/2} rceil , N/H is cyclic of arbitrarily large prime order and d(H,G)=n. We investigate the series d(H_n,G_n) in the cases when the depth, d(H_1,G_1), is 1, 2 or 3, where H_n:=H_1times G^{n-1} and G_n:=G_1wr C_n. We also prove that if H_1=S_k and G_1=S_{k+1} then d(H_n,G_n)=2nk-1.

Coset-wise affine functions and cycle types of complete mappings

Let K be a finite field of characteristic p. We study a certain class of functions K→K that agree with an Fp-affine function K→K on each coset of a given additive subgroup W of K – we call them W-coset-wiseFp-affine functions of K. We show that these functions form a permutation group on K with the structure of an imprimitive wreath product and characterize which of them are complete mappings of K. As a consequence, we are able to provide various new examples of cycle types of complete mappings of K – for instance, if p>2, then all cycle types where each cycle has length a power of p are achieved by complete mappings of K.

On the automorphism groups of certain wreath products

On the automorphism groups of certain wreath products

Some theorems on wreath products

Some theorems on wreath products

Permanence properties of verbal products and verbal wreath products of groups

By means of analyzing the notion of verbal products of groups, we show that soficity, hyperlinearity, amenability, the Haagerup property, the Kazhdan property (T), and exactness are preserved under taking k -nilpotent products of groups, while being orderable is not preserved. We also study these properties for solvable and for Burnside products of groups. We then show that if two discrete groups are sofic, or have the Haagerup property, their restricted verbal wreath product arising from nilpotent, solvable, and certain Burnside products is also sofic or has the Haagerup property, respectively. We also prove related results for hyperlinear, linear sofic, and weakly sofic approximations. Finally, we give applications combining our work with the Shmelkin embedding to show that certain quotients of free groups are sofic or have the Haagerup property.

Central automorphisms of standard wreath products

Central automorphisms of standard wreath products

Asynchronous wreath product and cascade decompositions for concurrent behaviours

We develop new algebraic tools to reason about concurrent behaviours modelled as languages of Mazurkiewicz traces and asynchronous automata. These tools reflect the distributed nature of traces and the underlying causality and concurrency between events, and can be said to support true concurrency. They generalize the tools that have been so efficient in understanding, classifying and reasoning about word languages. In particular, we introduce an asynchronous version of the wreath product operation and we describe the trace languages recognized by such products (the so-called asynchronous wreath product principle). We then propose a decomposition result for recognizable trace languages, analogous to the Krohn-Rhodes theorem, and we prove this decomposition result in the special case of acyclic architectures. Finally, we introduce and analyze two distributed automata-theoretic operations. One, the local cascade product, is a direct implementation of the asynchronous wreath product operation. The other, global cascade sequences, although conceptually and operationally similar to the local cascade product, translates to a more complex asynchronous implementation which uses the gossip automaton of Mukund and Sohoni. This leads to interesting applications to the characterization of trace languages definable in first-order logic: they are accepted by a restricted local cascade product of the gossip automaton and 2-state asynchronous reset automata, and also by a global cascade sequence of 2-state asynchronous reset automata. Over distributed alphabets for which the asynchronous Krohn-Rhodes theorem holds, a local cascade product of such automata is sufficient and this, in turn, leads to the identification of a simple temporal logic which is expressively complete for such alphabets.

On characters of wreath products

A character identity which relates irreducible character values of the hyperoctahedral group \(B_n\) to those of the symmetric group \(S_{2n}\) was recently proved by Lübeck and Prasad. Their proof is algebraic and involves Lie theory. We present a short combinatorial proof of this identity, as well as a generalization to other wreath products.Mathematics Subject Classifications: 20C30, 20E22, 05E10Keywords: Character identity, wreath product, partition, Murnaghan-Nakayama rule, colored permutations

Free wreath products with amalgamation

We define and study a notion of free wreath product with amalgamation for compact quantum groups. These objects were already introduced in the case of duals of discrete groups under the name “free wreath products of pairs” in a previous work of ours. We give several equivalent descriptions and use them to establish properties like residual finiteness, the Haagerup property or a smash product decomposition.

Jordan-like characterization of automorphism groups of planar graphs

We investigate automorphism groups of planar graphs. The main result is a complete recursive description of all abstract groups that can be realized as automorphism groups of planar graphs. The characterization is formulated in terms of inhomogeneous wreath products. In the proof, we combine techniques from combinatorics, group theory, and geometry. Our result significantly improves the Babai's description (1975).

Generalized Cyclotomic Mappings: Switching Between Polynomial, Cyclotomic, and Wreath Product Form

This paper is concerned with so-called index $d$ generalized cyclotomic mappings of a finite field $\mathbb{F}_q$, which are functions $\mathbb{F}_q\rightarrow\mathbb{F}_q$ that agree with a suitable monomial function $x\mapsto ax^r$ on each coset of the index $d$ subgroup of $\mathbb{F}_q^{\ast}$. We discuss two important rewriting procedures in the context of generalized cyclotomic mappings and present applications thereof that concern index $d$ generalized cyclotomic permutations of $\mathbb{F}_q$ and pertain to cycle structures, the classification of $(q-1)$-cycles and involutions, as well as inversion.

Affine wreath product algebras with trace maps of generic parity

The goal of this article is to study the structure and representation theory of affine wreath product algebras and its cyclotomic quotients These algebras appear naturally in Heisenberg categorification and generalize many important algebras (degenerate affine Hecke algebras, affine Sergeev algebras and wreath Hecke algebras). The whole class was introduced by D. Rosso and A. Savage in [16]. In [19], Savage studied both structure and representations under the condition that the trace map of A is even. In this paper we extend the definition for the case of odd trace. Since our approach is analogous to Savage’s, we consider the trace map being of arbitrary parity and unify statements and proofs. We also use an approach based on string diagrams, in the spirit of [5]. For simplicity of exposition, we assume A to be symmetric.

Full Transformation Semigroup of A Free Left S-Act on N-Generators

It is well known that the wreath product is the endmorphism monoid of a free S-act with n-generators. If S is a trivial semigroup then is isomorphic to . The extension for to where is an independent algebra has been investigated. In particular, we consider is to be , where is a free left S-act of n-generators. The eventual goal of this paper is to show that is an endomorphism monoid of a free left S-act of n-generators and to prove that is embedded in the wreath product .

Quantifying local embeddings into finite groups

We study a function LΓ which quantifies the LEF (local embeddability into finite groups) property for a finitely generated group Γ. We compute this LEF growth function in some examples, including certain wreath products. We compare LEF growth with the analogous quantitative version of residual finiteness, and exhibit a family of finitely generated residually finite groups which nevertheless admit many more local embeddings into finite groups than they do finite quotients. Along the way, we give a new proof that B.H. Neumann's continuous family of 2-generated groups contains no finitely presented group, a result originally due to Baumslag and Miller. We compare LΓ with quantitative versions of soficity and other metric approximation properties of groups. Finally, we show that there exists a “universal” function which is an upper bound on the LEF growth of any group on a given number of generators, and that (for non-cyclic groups) any such function is non-computable.

Investigating the solubility of wreath products group of degree 3p using numerical approach

Investigating the solubility of wreath products group of degree 3p using numerical approach