Orientable quadratic equations in wreath products of abelian groups

Abstract To each finitely generated group G , we associate a quasi-isometric invariant called the Dehn spectrum of G . If G is finitely presented, our invariant is closely related to the Dehn function of G , yet provides more information by encoding the isoperimetric behaviour of G at various scales. The main goal of this paper is to initiate the study of the Dehn spectrum of finitely generated (but not necessarily finitely presented) groups. In particular, we compute the Dehn spectrum of small cancellation groups, certain wreath products, and free Burnside groups of sufficiently large odd exponent. We also address some natural questions on the structure of the poset of Dehn spectra. As an application, we show that there exist $$2^{\aleph _0}$$ 2 ℵ 0 pairwise non-quasi-isometric finitely generated groups of finite exponent.

On wreath product occurring as subgroup of automata group

In this paper, we show that all Coleman automorphisms of wreath products of some finite groups with trivial centers by any finite group are inner. In particular, the normalizer property holds for these groups.

Abstract In this paper, we show that wreath products of groups have linear divergence, and we generalise the argument to permutational wreath products. We also prove that Houghton groups $${\mathcal {H}}_m$$ H m with $$m\ge 2$$ m ≥ 2 and Baumslag-Solitar groups have linear divergence. We explain how to generalise the argument for wreath products so that it holds for halo products of groups whose halo is large-scale commutative. Finally, we show that wreath products of graphs and Diestel-Leader graphs have linear divergence. The argument for Diestel-Leader graphs is further generalised to horocyclic products of proper, geodesically complete, Busemann $$\delta $$ δ -hyperbolic spaces that are uniformly not a quasi-line.

Throughout this paper our study focuses on transformation semigroups. These kinds of semigroups are the corner stone of semigroup theory. This is because every semigroup is isomorphic to transformation semigroup. The (total) transformation monoid on a finite set where , , is a semigroup of mapping that takes a set into itself, under the operation of composition of mapping with identity . In this paper, we use an algebraic method for considering the monoid , where an independence algebra is a disjointed union of sets of the form for all 1 Firstly, particular attention is paid to find the isomorphism between and the endomorphism monoid Secondly, the embeddedness of in (full) wreath product of by has been found. Finally, the description of Green's relation of has been provided.

We affirm several special cases of a conjecture that first appears in Alspach et al. (1987) which stipulates that the wreath (lexicographic) product of two hamiltonian decomposable di- rected graphs is also hamiltonian decomposable. Specifically, we show that the wreath product of hamiltonian decomposable directed graph G, such that |V (G)| is even and |V (G)| ⩾ 3, with a directed m-cycle such that m ⩾ 4 or the complete symmetric directed graph on m vertices such that m ⩾ 3, is hamiltonian decomposable. We also show the wreath product of a directed n-cycle, where n is even, with a directed m-cycle, where m ∈ {2, 3}, is not hamiltonian decomposable.

We develop general methods to compute the algebraic $K$-theory of crossed products by Bernoulli shifts on additive categories. From this we obtain a $K$-theory formula for regular group rings associated to wreath products of finite groups by groups satisfying the Farrell--Jones conjecture.

In this paper, we present generalizations of some results on the asymptotic property C for wreath products. Specifically, we prove that certain wreath-like products admit asymptotic property C, thus providing some new examples for further study.

A strong Gelfand pair (𝐺, 𝐻) consists of a group 𝐺 and its subgroup 𝐻, characterized by the property that the induced representation Ind 𝑑 𝐺 𝐻 of the permutation representation associated with the action of 𝐺 on the set of cosets 𝐺𝐻 is multiplicity-free. In this study, the conditions under which 𝐺 forms a strong Gelfand pair with certain subgroups are investigated, with emphasis on two main cases: i) the wreath product 𝐺 ≀ 𝑆 𝑛 with its subgroup 𝐺 ≀ 𝑆 𝑛−1 , ii) the group 𝑊 𝑛 = 𝐺 ≀ 𝑆 𝑛 = 𝐺 𝑛 ⋊ 𝑆 𝑛 with its subgroup 𝐶 𝑛 = 𝐸 𝑛 × 𝑆 𝑛 , where 𝐸 𝑛 is the diagonal subgroup of the direct product of 𝑛 copies of 𝐺. While such pairs are always Gelfand for abelian groups, in general they do not satisfy the strong Gelfand property. Specifically, there exists a positive integer 2 ≤ 𝑆𝑁(𝐺) < |𝐺| such that (𝐺 ≀ 𝑆 𝑛−1 , 𝐺 ≀ 𝑆 𝑛 ) and (𝑊 𝑛 , 𝐶 𝑛 ) are strong Gelfand pairs for 𝑛 < 𝑆𝑁(𝐺), but not for 𝑛 ≥ 𝑆𝑁(𝐺). This result holds for both abelian and non-abelian finite groups. A method is developed to compute this critical number 𝑆𝑁(𝐺), and explicit examples are provided for specific groups.

Abstract Let $G = X \wr H$ be the wreath product of a nontrivial finite group X with k conjugacy classes and a transitive permutation group H of degree n acting on the set of n direct factors of X n . If H is semiprimitive, then $k(G) \leq k^n$ for every sufficiently large n or k . This result solves a case of the non-coprime k ( GV ) problem and provides an affirmative answer to a question of Garzoni and Gill for semiprimitive permutation groups. The proof does not require the classification of finite simple groups.

Abstract Let 𝔼 be the HNN-extension of a group 𝐵 with subgroups 𝐻 and 𝐾 associated by an isomorphism φ : H → K \varphi\colon H\to K . Suppose that 𝐻 and 𝐾 are normal in 𝐵 and ( H ∩ K ) ⁢ φ = H ∩ K (H\cap K)\varphi=H\cap K . Under these assumptions, we prove necessary and sufficient conditions for 𝔼 to be residually a 𝒞-group, where 𝒞 is a class of groups closed under taking subgroups, quotient groups, and unrestricted wreath products. Among other things, these conditions give new facts on the residual finiteness and the residual 𝑝-finiteness of the group 𝔼.

Abstract Given a morphism $\varphi \;:\; G \to A \wr B$ from a finitely presented group G to a wreath product $A \wr B$ , we show that, if the image of $\varphi$ is a sufficiently large subgroup, then $\mathrm{ker}(\varphi)$ contains a non-abelian free subgroup and $\varphi$ factors through an acylindrically hyperbolic quotient of G. As direct applications, we classify the finitely presented subgroups in $A \wr B$ up to isomorphism and we deduce that a finitely presented group having a wreath product $(\text{non-trivial}) \wr (\text{infinite})$ as a quotient must be SQ-universal (extending theorems of Baumslag and Cornulier–Kar). Finally, we exploit our theorem in order to describe the structure of the automorphism groups of several families of wreath products, highlighting an interesting connection with the Kaplansky conjecture on units in group rings.

Abstract Starting with an integral domain D of characteristic 0, we consider a class of iterated wreath product $$W_n$$ of n copies of D. In order that $$W_n$$ be transfinite hypercentral, it is necessary to restrict to the case of wreath products defined by way of numerical polynomials. We also associate to each of these groups a Lie ring, providing a correspondence preserving most of the structure. This construction generalizes a result of Sushchansky and Netreba (Algebra Discrete Math 122–132, 2005) which characterizes the Lie algebras associated to the Sylow $$p$$ -subgroups of the symmetric group $${{\,\textrm{Sym}\,}}(p^n)$$ . As an application, we explore the normalizer chain $$\lbrace \textbf{N}_{i}\rbrace _{i\ge -1}$$ starting from the canonical regular abelian subgroup T of $$W_n$$ . Finally, we characterize the regular abelian normal subgroups of $$\textbf{N}_0$$ that are isomorphic to $$D^n$$ .

We consider the descent and flag major index statistics on the colored permutation groups, which are wreath products of the form $\mathfrak{S}_{n,r}=\mathbb{Z}_r\wr \mathfrak{S}_n$. We show that the $k$-th moments of these statistics on $\mathfrak{S}_{n,r}$ will coincide with the corresponding moments on all conjugacy classes without cycles of lengths $1,2,\ldots,2k$. Using this, we establish the asymptotic normality of the descent and flag major index statistics on conjugacy classes of $\mathfrak{S}_{n,r}$ with sufficiently long cycles. Our results generalize prior work of Fulman involving the descent and major index statistics on the symmetric group $\mathfrak{S}_n$. Our methods involve an intricate extension of Fulman's work on $\mathfrak{S}_n$ combined with the theory of the degree for a colored permutation statistic, as introduced by Campion Loth, Levet, Liu, Sundaram, and Yin.

In this paper, we consider wreath products of the form [Formula: see text] where [Formula: see text] is finite and simple. We compute word metrics in these groups and show that under these metrics the groups have the metric version LEF.

Let $R$ and $S$ be pomonoids and $_{R}{A}$ be a left $R$-poset. The wreath product of the pomonoids $R$ and $S$ by $_{R}{A}$ is defined as the pomonoid $T~=~R \times F(A, S)$ While, the wreath product $_TC$ of the left $R$-poset $_{R}{A}$ with the left $S$-poset $_{S}{B}$ over the pomonoid $T= R \times F(A, S)$ is the left $T$-poset ${_T C}= {_R A}\times {_S B}$ endowed with the monotone action given by $(r, f)(a, b) = (ra, f(a)b),$ where $(r, f) \in R\times F(A, S)$ and $ (a, b) \in A\times B$. The po-injectivity and po-cancellative properties on the wreath product $_TC$ are studied and the relations between them are established. The relation between po-surjective property and other properties on the wreath product $_TC$ are also established. Finally the characterization of some properties of po-flatness such as po-torsion free, properties $(P)$, $(E)$, $(P_E)$, and strongly flat have been examined on the wreath product $_TC$ and the relations among them have also been established.

Given a topologically free action of a countably infinite amenable group on the Cantor set, we prove that, for every subgroup G of the topological full group containing the alternating group, the group von Neumann algebra \mathscr{L}G is a McDuff factor. This yields the first examples of nonamenable simple finitely generated groups G for which \mathscr{L}G is McDuff. Using the same construction we show moreover that if a faithful action G\curvearrowright X of a countable group on a countable set with no finite orbits is amenable then the crossed product of the associated shift action over a given II _{1} factor is a McDuff factor. In particular, if H is a nontrivial countable ICC group and G\curvearrowright X is a faithful amenable action of a countable ICC group on a countable set with no finite orbits, then the group von Neumann algebra of the generalized wreath product H\wr_{X} G is a McDuff factor. Our technique can also be applied to show that if H is a nontrivial countable group and G\curvearrowright X is an amenable action of a countable group on a countable set with no finite orbits, then the generalized wreath product H\wr_{X} G is Jones–Schmidt stable.

Bases of twisted wreath products

Abstract Every word w in the free group F r of rank r induces a probability measure (the w -measure) on every compact group G , by substitution of Haarrandom G -elements in the letters. This measure is determined by its Fourier coefficients: the w -expectations $${\mathbb E}_{w}[\chi]$$ E w [ χ ] of the irreducible characters of G . For every compact group G , the wreath product with the symmetric group G ≀ S n has some natural irreducible characters χ , and we approximate $${\mathbb E}_{w}[\chi]$$ E w [ χ ] for every word w ∈ F r , revealing new automorphism-invariant quantities of words that generalize the primitivity rank π ( w ). This generalizes previous works by Puder–Parzanchevski and Magee–Puder. We demonstrate applications to automorphism groups of trees, investigate properties of the new invariants, and show polynomial decay of $${\mathbb E}_{w}[\chi]$$ E w [ χ ] also for wreath products with more general actions.