Related papers: Enumerating Palindromes and Primitives in Rank Two…

The present paper records more details of the relationship between primitive elements and palindromes in F_2, the free group of rank two. We characterise the conjugacy classes of primitive elements which contain palindromes as those which…

We consider non-elementary representations of two generator free groups in $PSL(2,\mathbb{C})$, not necessarily discrete or free, $G = < A, B >$. A word in $A$ and $B$, $W(A,B)$, is a palindrome if it reads the same forwards and backwards.…

A palindrome is a word that reads the same left-to-right as right-to-left. We show that every simple group has a finite generating set $X$, such that every element of it can be written as a palindrome in the letters of $X$. Moreover, every…

Given a (finite or infinite) subset $X$ of the free monoid $A^*$ over a finite alphabet $A$, the rank of $X$ is the minimal cardinality of a set $F$ such that $X \subseteq F^*$. A submonoid $M$ generated by $k$ elements of $A^*$ is…

Given a (finite or infinite) subset $X$ of the free monoid $A^*$ over a finite alphabet $A$, the rank of $X$ is the minimal cardinality of a set $F$ such that $X \subseteq F^*$. We say that a submonoid $M$ generated by $k$ elements of $A^*$…

The Gilman-Maskit algorithm for determining the discreteness or non-discreteness of a two-generator subgroup of $\mathrm{PSL}_2\mathbb{R}$ terminates with a pair of generators that are Farey words. The Farey words are primitive words that…

The palindromic length of a finite word $w$ is defined as the minimal number of palindromes such that their product is $w$. Clearly, this function may take different values depending on if we consider $w$ as an element a free semigroup or…

In the free group $F_k$, an element is said to be primitive if it belongs to a free generating set. In this paper, we describe what a generic primitive element looks like. We prove that up to conjugation, a random primitive word of length…

The centrepiece of this paper is a normal form for primitive elements which facilitates the use of induction arguments to prove properties of primitive elements. The normal form arises from an elementary algorithm for constructing a…

Let G be a group and S a subset of G that generates G. For each x in G define the length l_S(x) of x relative to S to be the minimal k such that x is a product of k elements of S. The supremum of the values l_S(x), x \in G, is called the…

Let $F_k$ be the free group on $k$ generators, and let $H\le J\le \F_k$ be subgroups of finite rank. We present a new elementary algorithm to determine whether $H$ is a free factor of $J$. In particular, this algorithm can determine whether…

Let F_k be the free group on k generators. A word w \in F_k is called primitive if it belongs to some basis of F_k. We investigate two criteria for primitivity, and consider more generally, subgroups of F_k which are free factors. The first…

We establish new characterizations of primitive elements and free factors in free groups, which are based on the distributions they induce on finite groups. For every finite group $G$, a word $w$ in the free group on $k$ generators induces…

We introduce two classes of morphisms over the alphabet $A=\{0,1\}$ whose fixed points contain infinitely many antipalindromic factors. An antipalindrome is a finite word invariant under the action of the antimorphism…

In a 1995 paper, Hof, Knill and Simon obtain a sufficient combinatorial criterion on the hull $\Omega$ of the potential of a discrete Schr\"odinger operator which guarantees purely singular continuous spectrum on a generic subset of…

For a free group $F_r$ of finite rank $r\ge 2$ and a nontrivial element $w\in F_r$ the \emph{primitivity rank} $\pi(w)$ is the smallest rank of a subgroup $H\le F_r$ such that $w\in H$ and that $w$ is not primitive in $H$ (if no such $H$…

A palindrome in a free group F_n is a word on some fixed free basis of F_n that reads the same backwards as forwards. The palindromic automorphism group \Pi A_n of the free group F_n consists of automorphisms that take each member of some…

We prove that if $L=\mbox{}^2F_4(2^{2n+1})'$ and $x$ is a nonidentity automorphism of $L$ then $G=\langle L,x\rangle$ has four elements conjugate to $x$ that generate $G$. This result is used to study the following conjecture about the…

An element w of the extension E of degree n over the finite field F=GF(q) is called free over F if {w, w^q,...,w^{q^{n-1}}} is a (normal) basis of E/F. The Primitive Normal Basis Theorem, first established in full by Lenstra and Schoof…

Palindromes are those reduced words of free products of groups that coincide with their reverse words. We prove that a free product of groups $G$ has infinite palindromic width, provided that $G$ is not the free product of two cyclic groups…