相关论文: On certain C-test words for free groups

Let F_n be the free group of rank n, with generating set S=\{x_1,...,x_n\}. An automorphism \phi of F_n is called symmetric if for each 1\leq i\leq n, \phi(x_i) is conjugate to x_j or x_j^{-1} for some 1\leq j\leq n. Let \Sigma Aut(F_n) be…

Let $\gamma_n=[x_1,\dots,x_n]$ be the $n$th lower central word. Denote by $X_n$ the set of $\gamma_n$-values in a group $G$ and suppose that there is a number $m$ such that $|g^{X_n}|\leq m$ for each $g\in G$. We prove that…

A binary word is a map W : N --> {0,1}, and the set of factors of W with length n is F_n(W):={(W(i),W(i+1),...,W(i+n-1)) : i >= 0}. A word is Sturmian if |F_n(W)|=n+1 for every n>0. We show that the sum of the heights (also known as hamming…

In this article, we study geometric properties of nilpotent groups. We find a geometric criterion for the word problem for the finitely generated free nilpotent groups. By geometric criterion, we mean a way to determine whether two words…

Let $F_n$ be the free group of a finite rank $n$. We study orbits $Orb_{\phi}(u)$, where $u$ is an element of the group $F_n$, under the action of an automorphism $\phi$. If an orbit like that is finite, we determine precisely what its…

Consider a random word $X^n=(X_1,\ldots ,X_n)$ in an alphabet consisting of $4$ letters, with the letters viewed either as $A$, $U$, $G$ and $C$ (i.e., nucleotides in an RNA sequence) or $\alpha$, $\bar{\alpha}$, $\beta$ and $\bar{\beta}$…

Given a bi-order $\succ$ on the free group $\mathcal{F}$, we show that every non-periodic cyclically reduced word $W\in \mathcal{F}$ admits a maximal ascent that is uniquely positioned. This provides a cyclic permutation of $W'$ that…

We find a lower bound to the size of finite groups detecting a given word in the free group, more precisely we construct a word w_n of length n in non-abelian free groups with the property that w_n is the identity on all finite quotients of…

Howie and Duncan observed that a word in a free product with length at least two and which is not a proper power can be decomposed as a product of two cyclic subwords each of which is uniquely positioned. Using this property, they proved…

A finite word $w$ of length $n$ contains at most $n+1$ distinct palindromic factors. If the bound $n+1$ is attained, the word $w$ is called rich. An infinite word $w$ is called rich if every finite factor of $w$ is rich. Let $w$ be a word…

We introduce and study a complexity function on words $c_x(n),$ called \emph{cyclic complexity}, which counts the number of conjugacy classes of factors of length $n$ of an infinite word $x.$ We extend the well-known Morse-Hedlund theorem…

A group word $w$ is said to be strongly concise in a class $\mathscr C$ of profinite groups if, for any group $G$ in $\mathscr C$, either $w$ takes at least continuum values in $G$ or the verbal subgroup $w(G)$ is finite. It is conjectured…

We generalize the combinatorial approaches of Rapaport and Higgins--Lyndon to the Whitehead algorithm. We show that for every automorphism $\varphi$ of a free group $F$ and every word $u\in F$ there exists a finite multiset of words…

We consider words over a binary alphabet. A word $w$ is overlap-free if it does not have factors (blocks of consecutive letters) of the form $uvuvu$ for nonempty $u$. Let $M(w)$ denote the number of positions that are middle positions of…

Let v and w be nontrivial words in two free groups. We prove that, for all sufficiently large finite non-abelian simple groups G, there exist subsets C of v(G) and D of w(G) of size such that every element of G can be realized in at least…

Let m, n be positive integers, v a multilinear commutator word and w = v^m. Denote by v(G) and w(G) the verbal subgroups of a group G corresponding to v and w, respectively. We prove that the class of all groups G in which the w-values are…

Elements of the commutator subgroup of a free group can be presented as values of canonical forms, called Wicks forms. We show that, starting from sufficiently high genus g, there is a sequence of words w(g) which can be presented by f(g)…

Using the theory developed by Olga Kharlampovich, Alexei Miasnikov, and, independently, by Zlil Sela to describe the set of homomorphisms of a f.g. group G into a free group F, we describe the solutions to equations with coefficients from F…

A word $w$ is concise in a class of groups $\mathcal{C}$ if, for every group $G$ in $\mathcal{C}$, the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in $G$. This notion can be naturally extended to…

If $x$ is a non-empty string then the repetition $xx$ is called a tandem repeat. Similarly, a tandem in a two dimensional array $X$ is a configuration consisting of a same primitive block $W$ that touch each other with one side or corner.…