Narrow Results

We prove that any countable sofic (respectively, weakly sofic, hyperlinear, linearly sofic) group can be embedded into a finitely generated sofic (respectively, weakly sofic, hyperlinear, linearly sofic) group. We also prove an analogous result that any countable dimensional linearly sofic Lie algebra can be embedded into a finitely generated linearly sofic Lie algebra. In the course of proving this result, we prove that, over a field of characteristic 0, every extension of a linearly sofic Lie algebra by a Lie algebra with amenable universal enveloping algebra is linearly sofic.

The purpose of this paper is to introduce the notion of several kinds of products of rough transformation semigroups. We establish relationship through coverings and investigate some algebraic properties with regard to these products.

Inspired by the reconstruction program of conformal field theories of Vaughan Jones we recently introduced a vast class of the so-called forest-skein groups. They are built from a skein presentation: a set of colors and a set of pairs of colored trees. Each nice skein presentation produces four groups similar to Richard Thompson’s group $F,T,V$ and the braided version $BV$ of Brin and Dehornoy.

In this paper, we consider forest-skein groups obtained from one-dimensional skein presentations; the data of a homogeneous monoid presentation. We decompose these groups as wreath products. This permits to classify them up to isomorphisms. Moreover, we prove that a number of properties of the fraction group of the monoid pass through the forest-skein groups such as the Haagerup property, homological and topological finiteness properties, and orderability.

We extend the tree-wreath product of groups introduced by Brunner and the author, as a generalization of the restricted wreath product, thus enlarging the class of groups known to be generated by finite synchronous automata. In particular, we prove that given a countable abelian residually finite 2 -group H and B = B(n,ℤ), a canonical subgroup of finite index in GL(n,ℤ), then the restricted wreath product HwrB can be generated by finite synchronous automata on 0,1. This is obtained by producing a representation of B as a group of automorphisms of the binary tree such that the stabilizer of the infinite sequence of 0's is trivial. The uni-triangular group U = U(n,ℤ) is a subgroup of B(n,ℤ) and so, HwrU also can be generated by finite synchronous automata on 0,1.

Let L≀X be a lamplighter graph, i.e., the graph-analogue of a wreath product of groups, and let P be the transition operator (matrix) of a random walk on that structure. We explain how methods developed by Saloff-Coste and the author can be applied for determining the ℓ^p-norms and spectral radii of P, if one has an amenable (not necessarily discrete or unimodular) locally compact group of isometries that acts transitively on L. This applies, in particular, to wreath products K≀G of finitely-generated groups, where K is amenable. As a special case, this comprises a result of Żuk regarding the ℓ²-spectral radius of symmetric random walks on such groups.

We show that the probability of generating an iterated wreath product of non-abelian finite simple groups converges to 1 as the order of the first simple group tends to infinity provided the wreath products are constructed with transitive and faithful actions. This has the consequence that the profinite group which is the inverse limit of these iterated wreath products is positively finitely generated.

This paper introduces the notion of a module over a graph and defines the wreath product and derived module of a relational morphism in this context.

The inverse semigroup of partial automaton permutations over a finite alphabet is characterized in terms of wreath products. The permutation conjugacy relation in this semigroup and the Green's relations are described. Criteria of primary conjugacy and conjugacy are given for certain naturally defined families of partial automaton permutations. Sufficient conditions under which an inverse semigroup admits a level transitive action are presented. We give explicit examples (monogenic inverse semigroups and some commutative Clifford semigroups) of inverse semigroups generated by finite automata.

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.

In this paper we combine the algebraic properties of Mealy machines generating self-similar groups and the combinatorial properties of the corresponding deterministic finite automata (DFA). In particular, we relate bounded automata to finitely generated synchronizing automata and characterize finite automata groups in terms of nilpotency of the corresponding DFA. Moreover, we present a decidable sufficient condition to have free semigroups in an automaton group. A series of examples and applications is widely discussed, in particular we show a way to color the de Bruijn automata into Mealy automata whose associated semigroups are free, and we present some structural results related to the associated groups.

In contrast to being automatic, being Cayley automatic a priori has no geometric consequences. Specifically, Cayley graphs of automatic groups enjoy a fellow traveler property. Here, we study a distance function introduced by the first author and Trakuldit which aims to measure how far a Cayley automatic group is from being automatic, in terms of how badly the Cayley graph fails the fellow traveler property. The first author and Trakuldit showed that if it fails by at most a constant amount, then the group is in fact automatic. In this paper, we show that for a large class of non-automatic Cayley automatic groups this function is bounded below by a linear function in a precise sense defined herein. In fact, for all Cayley automatic groups which have super-quadratic Dehn function, or which are not finitely presented, we can construct a non-decreasing function which (1) depends only on the group and (2) bounds from below the distance function for any Cayley automatic structure on the group.

We prove that the set of subgroups of the automorphism group of a two-sided full shift is closed under countable graph products. We introduce the notion of a group action without A-cancellation (for an abelian group A), and show that when A is a finite abelian group and G is a group of cellular automata whose action does not have A-cancellation, the wreath product $A\wr G$ embeds in the automorphism group of a full shift. We show that all free abelian groups and free groups admit such cellular automata actions. In the one-sided case, we prove variants of these results with reasonable alphabet blow-ups.

Given a knot $K$, we may construct a group $G_n(K)$ from the fundamental group of $K$ by adjoining an $n$th root of the meridian that commutes with the corresponding longitude. For $n\ge 2$ these “generalized knot groups” determine $K$ up to reflection.

The second author has shown that for $n\ge 2$, the generalized knot groups of the square and granny knots can be distinguished by counting homomorphisms into a suitably chosen finite group. We extend this result to certain generalized knot groups of square and granny knot analogues $S{K}_{a,b}=T_{a,b}\#T_{-a,b}$, $G{K}_{a,b}=T_{a,b}\#T_{a,b}$, constructed as connected sums of $(a,b)$-torus knots of opposite or identical chiralities. More precisely, for coprime $a,b\ge 2$ and $n$ satisfying a coprimality condition with $a$ and $b$, we construct an explicit finite group $G$ (depending on $a$, $b$ and $n$) such that $G_n(S{K}_{a,b})$ and $G_n(G{K}_{a,b})$ can be distinguished by counting homomorphisms into $G$. The coprimality condition includes all $n\ge 2$ coprime to $ab$. The result shows that the difference between these two groups can be detected using a finite group.

In this paper, Rubik’s cube group in $n$-dimensional case is described. In that case, some interesting effects arise, which is related with angular rotations. For $n=4$, the group ${\mathbb{Z}}_3$ arises, which can be realized as a factor group of $A_4$ by commutation $[A_4,A_4]=V_4$. For $n\ge 5$ there are no such effects, because corresponding factor is trivial. Rubik’s cube is an important model to study effects of groups related with, in particular, holonomy groups from differential geometry.

We compute the upper central series for the regular wreath product finite group C ≀ E where C is a cyclic p-group and E is an elementary abelian p-group for some prime p. The notion of a pattern subgroup enables us to describe the upper central series in a natural way.

We study the 2-adic properties for the numbers of involutions in the alternative groups, and give an affirmative answer to a conjecture of Kim and Kim [A combinatorial approach to the power of 2 in the number of involutions, J. Combin. Theory Ser. A117 (2010) 1082–1094]. Some analogous and general results are also presented.

In this paper, we shall introduce thin n-subpolygroups of a given n-polygroup and in this regards, the notion of wreath product of n-polygroups will be studied. Also, double cosets of n-polygroups are investigated and the classical isomorphism theorems of groups are generalized to n-polygroups. The main result of the paper is that a finite n-polygroup is singular if and only if it is a wreath product of n-subpolygroups all of which are thin or generated by an involution or by an idempotent element.

A diagonal base of a Sylow 2-subgroup $P_n(2)$ of symmetric group $S_{2^n}$ is a minimal generating set of this subgroup consisting of elements with only one nonzero coordinate in the polynomial representation. For different diagonal bases, Cayley graphs over $P_n(2)$ may have different girths (i.e. minimal lengths of cycles). In this paper, all possible values of girths of Cayley graphs over $P_n(2)$ with diagonal bases are calculated. A criterion for whenever such Cayley graph has girth equal to 4 is presented.

A strong Gelfand pair is a pair $(G,H),H\le G$, of finite groups such that the Schur ring determined by the $H$-classes $g^H,g\in G$, is a commutative ring. We find all strong Gelfand pairs $(S_n,H)$. We also define an extra strong Gelfand pair $(G,H)$, this being a strong Gelfand pair of maximal dimension, and show that in this case $H$ must be abelian.

A subgroup $H$ of a group $G$ is said to be pronormal in $G$ if $H$ and $H^g$ are conjugate in $〈H,H^g〉$ for each $g\in G$. Some problems in Finite Group Theory, Combinatorics and Permutation Group Theory were solved in terms of pronormality, therefore, the question of pronormality of a given subgroup in a given group is of interest. Subgroups of odd index in finite groups satisfy a native necessary condition of pronormality. In this paper, we continue investigations on pronormality of subgroups of odd index and consider the pronormality question for subgroups of odd index in some direct products of finite groups. In particular, in this paper, we prove that the subgroups of odd index are pronormal in the direct product $G$ of finite simple symplectic groups over fields of odd characteristics if and only if the subgroups of odd index are pronormal in each direct factor of $G$. Moreover, deciding the pronormality of a given subgroup of odd index in the direct product of simple symplectic groups over fields of odd characteristics is reducible to deciding the pronormality of some subgroup $H$ of odd index in a subgroup of ${\prod }_{i=1}^t{\mathbb{Z}}_3\wr {\mathrm{\text{Sym}}}_{n_i}$, where each ${\mathrm{\text{Sym}}}_{n_i}$ acts naturally on $\{1,\dots ,n_i\}$, such that $H$ projects onto ${\prod }_{i=1}^t{\mathrm{\text{Sym}}}_{n_i}$. Thus, in this paper, we obtain a criterion of pronormality of a subgroup $H$ of odd index in a subgroup of ${\prod }_{i=1}^t{\mathbb{Z}}_{p_i}\wr {\mathrm{\text{Sym}}}_{n_i}$, where each $p_i$ is a prime and each ${\mathrm{\text{Sym}}}_{n_i}$ acts naturally on $\{1,\dots ,n_i\}$, such that $H$ projects onto ${\prod }_{i=1}^t{\mathrm{\text{Sym}}}_{n_i}$.