Palindromic Primitives and Palindromic Bases in the Free Group of Rank Two

The present paper records more details of the relationship between primitive elements and palindromes in F2, the free group of rank two. We characterize the conjugacy classes of palindromic primitive elements as those in which cyclically reduced words have odd length. We identify large palindromic subwords of certain primitives in conjugacy classes which contain cyclically reduced words of even length. We show that under obvious conditions on exponent sums, pairs of palindromic primitives form palindromic bases for F2. Further, we note that each cyclically reduced primitive element is either a palindrome, or the concatenation of two palindromes. Notation 0.1. For each natural number n ≥ 2, let Fn denote the nonabelian free group of rank n, which we identify with the set of reduced words in the alphabet An := {x1, . . . , xn}±. For elements w, v ∈ Fn, we write w ≡n v if w and v are equal words, and w =n v if w and v are equal elements of Fn. We write wv (w.c.) for the concatenation of the words w and v and wv for the product of w and v in Fn. We write |w|n for the word-length of w in An. Let Ψn : Fn → Fn be the map which reverses each word in Fn. For convenience, we usually omit the subscript n from ≡n, =n, Ψn, |·|n, and we write x := x1 and y := y1 (so F2 is the free group on two generators x and y). Recall that an element w ∈ Fn is said to be a palindrome if Ψ(w) = w (that is, “w reads the same forwards and backwards”) and primitive if it is an element of some basis for Fn. Much is known about the structure of primitive elements in F2 (see, for example, [3], [7], [8], [4], [1]) and indeed primitive elements in free groups of rank greater than two (see, for example, [5, pp.162-169], [2], [6], [1]). A newly emerging theme in the study of primitive elements in free groups is the relationship between primitive elements and palindromes (see [4], [1]). The present paper records more details of this relationship. The author is grateful to Peter Nickolas for a discussion which began this paper. This work was undertaken in the employment of The University of Wollongong, Australia.