Square free words as products of commutators

Products of Commutators and Products of Squares in a Free Group

A classification of the ways in which an element of a free group can be expressed as a product of commutators or as a product of squares is given. This is then applied to some particular classes of elements. Finally, a question about expressing a commutator as a product of squares is addressed.

Words and pronilpotent subgroups in profinite groups

Let $w$ be a multilinear commutator word, that is, a commutator of weight $n$ in $n$ different group variables. It is proved that if $G$ is a profinite group in which all pronilpotent subgroups generated by $w$-values are periodic, then the verbal subgroup $w(G)$ is locally finite.

Commutators and squares in free groups

Let F_2 be the free group generated by x and y. In this article, we prove that the commutator of x^m and y^n is a product of two squares if and only if mn is even. We also show using topological methods that there are infinitely many obstructions for an element in F_2 to be a product of two squares.

Profinite groups with restricted centralizers of commutators

A group G has restricted centralizers if for each g in G the centralizer C_G(g) either is finite or has finite index in G. A theorem of Shalev states that a profinite group with restricted centralizers is abelian-by-finite. In the present article we handle profinite groups with restricted centralizers of word-values. We show that if w is a multilinear commutator word and G a profinite group with restricted centralizers of w-values, then the verbal subgroup w(G) is abelian-by-finite.

Coverings of commutators in profinite groups

Let $w$ be a group-word. Suppose that the set of all $w$-values in a profinite group $G$ is contained in a union of countably many subgroups. It is natural to ask in what way the structure of the verbal subgroup $w(G)$ depends on the properties of the covering subgroups. The present article is a survey of recent results related to that question. In particular we survey results on finite and countable coverings of word-values (mostly commutators) by procyclic, abelian, nilpotent, and soluble subgroups, as well as subgroups with finiteness conditions. The last section of the paper is devoted to …

Finitely generated infinite simple groups of infinite square width and vanishing stable commutator length

It is shown that there exist finitely generated infinite simple groups of infinite commutator width and infinite square width on which there exists no stably unbounded conjugation-invariant norm, and in particular stable commutator length vanishes. Moreover, a recursive presentation of such a group with decidable word and conjugacy problems is constructed.

Computing commutator length is hard

The commutator length $cl_G(g)$ of an element $g \in [G,G]$ in the commutator subgroup of a group $G$ is the least number of commutators needed to express $g$ as their product. If $G$ is a non-abelian free groups, then given an integer $n \in \mathbb{N}$ and an element $g \in [G,G]$ the decision problem which determines if $cl_G(g) \leq n$ is NP-complete. Thus, unless P=NP, there is no algorithm that computes $cl_G(g)$ in polynomial time in terms of $|g|$, the wordlength of $g$. This statement remains true for groups which have a retract to a non-abelian free group, such as non-abelian right-a…

On commutator length in free groups

Let $F$ be a free group. We present for arbitrary $g\in\mathbb{N}$ a LogSpace (and thus polynomial time) algorithm that determines whether a given $w\in F$ is a product of at most $g$ commutators; and more generally an algorithm that determines, given $w\in F$, the minimal $g$ such that $w$ may be written as a product of $g$ commutators (and returns $\infty$ if no such $g$ exists). The algorithm also returns words $x_1,y_1,\dots,x_g,y_g$ such that $w=[x_1,y_1]\cdots[x_g,y_g]$. The algorithms we present are also efficient in practice. Using them, we produce the first example of a word in the fr…

Conciseness of coprime commutators in finite groups

Let $G$ be a finite group. We show that the order of the subgroup generated by coprime $γ_k$-commutators (respectively $δ_k$-commutators) is bounded in terms of the size of the set of coprime $γ_k$-commutators (respectively $δ_k$-commutators). This is in parallel with the classical theorem due to Turner-Smith that the words $γ_k$ and $δ_k$ are concise.

On groups admitting a word whose values are Engel

Let m, n be positive integers, v a multilinear commutator word and w = v^m. We prove that if G is a residually finite group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent. We also examine the question whether this is true in the case where G is locally graded rather than residually finite. We answer the question affirmatively in the case where m = 1. Moreover, we show that if u is a non-commutator word and G is a locally graded group in which all u-values are n-Engel, then the verbal subgroup u(G) is locally nilpotent.

On surjectivity of word maps on $\mathrm{PSL}_2$

Let $w = [[x^k, y^l], [x^m, y^n]]$ be a non-trivial double commutator word. We show that $w$ is surjective on $\mathrm{PSL}_2(K)$, where $K$ is an algebraically closed field of characteristic $0$.

On finite groups in which commutators are covered by Engel subgroups

Let $m,n$ be positive integers and $w$ a multilinear commutator word. Assume that $G$ is a finite group having subgroups $G_1,\ldots,G_m$ whose union contains all $w$-values in $G$. Assume further that all elements of the subgroups $G_1,\ldots,G_m$ are $n$-Engel in $G$. It is shown that the verbal subgroup $w(G)$ is $s$-Engel for some $\{m,n,w\}$-bounded number $s$.

Word Measures on Unitary Groups

We combine concepts from random matrix theory and free probability together with ideas from the theory of commutator length in groups and maps from surfaces, and establish new connections between the two. More particularly, we study measures induced by free words on the unitary groups $U(n)$. Every word $w$ in the free group $F_r$ on $r$ generators determines a word map from $U(n)^r$ to $U(n)$, defined by substitutions. The $w$-measure on $U(n)$ is defined as the pushforward via this word map of the Haar measure on $U(n)^r$. Let $Tr_w(n)$ denote the expected trace of a random unitary matrix sa…

Orientable and non-orientable genus $n$ Wicks forms over hyperbolic groups

In 1962 M.J. Wicks gave a precise description of the form a commutator could take in a free group or a free product and in 1973 extended this description to cover a product of two squares. Subsequently, lists of "Wicks forms" were found for arbitrary products of commutators and squares in free groups and free products, by Culler, Vdovina and other authors. Here we construct Wicks forms for products of commutators and squares in a hyperbolic group. As applications we give explicit lists of forms for a commutator and for a square, and find bounds on the lengths of conjugating elements required t…

关系子的稳定换位子长度并非单关系群不变量

The stable commutator length of a relator is not a one-relator group invariant

该研究通过构造两个同构的单关系群，证明关系子的稳定换位子长度并非单关系群不变量，否定了Heuer与Löh提出的相关问题。

On locally graded groups with a word whose values are Engel

Let m, n be positive integers, v a multilinear commutator word and w = v^m. We prove that if G is a locally graded group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent.

Conjugation-invariant norms on the commutator subgroup of the infinite braid group

In this paper, we give a proof of the result of Brandenbursky and Kȩdra which says that the commutator subgroup of the infinite braid group admits stably unbounded norms. Moreover, we observe the norms which we constructed are equivalent to the biinvariant word norm studied by Brandenbursky and Kȩdra.

The commutator and centralizer description of Sylow 2-subgroups of alternating and symmetric groups

Given a permutational wreath product sequence of cyclic groups of prime order we research a commutator width of such groups and some properties of its commutator subgroup. Commutator width of Sylow 2-subgroups of alternating group $A_{2^{k}}$, permutation group $S_{2^k}$ and $C_p \wr B$ were founded. The result of research was extended on subgroups $(Syl_2 {A_{2^k}})'$, $p>2$. The paper presents a construction of commutator subgroup of Sylow 2-subgroups of symmetric and alternating groups. Also minimal generic sets of Sylow 2-subgroups of $A_{2^k}$ were founded. Elements presentation of $(Syl_…

Surface Words are Determined by Word Measures on Groups

Every word $w$ in a free group naturally induces a probability measure on every compact group $G$. For example, if $w=\left[x,y\right]$ is the commutator word, a random element sampled by the $w$-measure is given by the commutator $\left[g,h\right]$ of two independent, Haar-random elements of $G$. Back in 1896, Frobenius showed that if $G$ is a finite group and $ψ$ an irreducible character, then the expected value of $ψ\left(\left[g,h\right]\right)$ is $\frac{1}{ψ\left(e\right)}$. This is true for any compact group, and completely determines the $\left[x,y\right]$-measure on these groups. An a…

Large scale geometry of commutator subgroups

Let G be a finitely presented group, and G' its commutator subgroup. Let C be the Cayley graph of G' with all commutators in G as generators. Then C is large scale simply connected. Furthermore, if G is a torsion-free nonelementary word-hyperbolic group, C is one-ended. Hence (in this case), the asymptotic dimension of C is at least 2.

Conjugacy growth of commutators

For the free group $F_r$ on $r>1$ generators (respectively, the free product $G_1 * G_2$ of two nontrivial finite groups $G_1$ and $G_2$), we obtain the asymptotic for the number of conjugacy classes of commutators in $F_r$ (respectively, $G_1 * G_2$) with a given word length in a fixed set of free generators (respectively, the set of generators given by the nontrivial elements of $G_1$ and $G_2$). Our result is proven by using the classification of commutators in free groups and in free products by Wicks, and builds on the works of Rivin and Sharp, who asymptotically counted the conjugacy cla…

On profinite groups with commutators covered by countably many cosets

Let w be a group-word. Suppose that the set of all w-values in a profinite group G is contained in a union of countably many cosets of subgroups. We are concerned with the question to what extent the structure of the verbal subgroup w(G) depends on the properties of the subgroups. We prove the following theorem. Let C be a class of groups closed under taking subgroups, quotients, and such that in any group the product of finitely many normal C-subgroups is again a C-subgroup. If w is a multilinear commutator and G is a profinite group such that the set of all w-values is contained in a union o…

On finiteness of verbal subgroups

Given a group-word $w$ and a group $G$, the set of $w$-values in $G$ is denoted by $G_w$ and the verbal subgroup $w(G)$ is the one generated by $G_w$. The word $w$ is concise if $w(G)$ is finite for all groups $G$ in which $G_w$ is finite. We obtain several results supporting the conjecture that the word $[u_1,\dots,u_s]$ is concise whenever the words $u_1,\dots,u_s$ are non-commutator.

On profinite groups in which commutators are covered by finitely many subgroups

For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite exponent $e$ whose union contains all $γ_k$-values in $G$, it is shown that $γ_k(G)$ has finite $(e,k,s)$-bounded exponent. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite rank $r$ whose union contain…

On the Width of Verbal Subgroups of the Groups of Triangular Matrices over a Field of Arbitrary Characteristic

The width $\wid(G,W)$ of the verbal subgroup $v(G,W)$ of a group $G$ defined by a collection of group words $W$ is the smallest number $m$ in $\mathbb N \cup {+\infty}$ such that every element of $v(G,W)$ is can be represented as the product of at most $m$ words in $W$ evaluated on the group $G$ and their inverses. Recall that every verbal subgroup of the group $T_{n} (K)$ of triangular matrices over an arbitrary field $K$ can be defined by just one word: an outer commutator word or a power word. We prove that for every outer commutator word $w$ the equality $\wid(T_{n} (K),w)=1$ holds on the …

The language of geodesics for the discrete Heisenberg group

In this paper, we give a complete description of the language of geodesic words for the discrete Heisenberg group $H(\mathbb{Z})$ with respect to the standard two-element generating set. More precisely, we prove that the only dead end elements in $H(\mathbb{Z})$ are nontrivial elements of the commutator subgroup. We give a description of their geodesic representatives, which are called dead end words. The description is based on a minimal perimeter polyomino concept. Finally, we prove that any geodesic word in $H(\mathbb{Z})$ is a prefix of a dead end word.

On countable coverings of word values in profinite groups

We prove the following results. Let w be a multilinear commutator word. If G is a profinite group in which all w-values are contained in a union of countably many periodic subgroups, then the verbal subgroup w(G) is locally finite. If G is a profinite group in which all w-values are contained in a union of countably many subgroups of finite rank, then the verbal subgroup w(G) has finite rank as well. As a by-product of the techniques developed in the paper we also prove that if G is a virtually soluble profinite group in which all w-values have finite order, then w(G) is locally finite and has…

Commutator Subgroups of Twin Groups and Grothendieck's Cartographical Groups

Let $TW_n$ be the twin group on $n$ arcs, $n \geq 2$. The group $TW_{m+2}$ is isomorphic to Grothendieck's $m$-dimensional cartographical group $\mathcal C_m$, $m \geq 1$. In this paper we give a finite presentation for the commutator subgroup $TW_{m+2}'$, and prove that $TW_{m+2}'$ has rank $2m-1$. We derive that $TW_{m+2}'$ is free if and only if $m \leq 3$. From this it follows that $TW_{m+2}$ is word-hyperbolic and does not contain a surface group if and only if $m \leq 3$. It also follows that the automorphism group of $TW_{m+2}$ is finitely presented for $m \leq 3$.

On the Fourier Expansion of Word Maps

Frobenius observed that the number of times an element of a finite group is obtained as a commutator is given by a specific combination of the irreducible characters of the group. More generally, for any word w the number of times an element is obtained by substitution in w is a class function. Thus, it has a presentation as a combination of irreducible characters, called its Fourier expansion. In this paper we present formulas regarding the Fourier expansion of words in which some letters appear twice. These formulas give simple proofs for classical results, as well as new ones.

The word problem for some uncountable groups given by countable words

We investigate the fundamental group of Griffiths' space, and the first singular homology group of this space and of the Hawaiian Earring by using (countable) reduced tame words. We prove that two such words represent the same element in the corresponding group if and only if they can be carried to the same tame word by a finite number of word transformations from a given list. This enables us to construct elements with special properties in these groups. By applying this method we prove that the two homology groups contain uncountably many different elements that can be represented by infinit…