Enumerating Palindromes and Primitives in Rank Two Free Groups

Let F = 〈a, b〉 be a rank two free group. A word W (a, b) in F is primitive if it, along with another group element, generates the group. It is a palindrome (with respect to a and b) if it reads the same forwards and backwards. It is known that in a rank two free group any primitive element is conjugate either to a palindrome or to the product of two palindromes, but known iteration schemes for all primitive words give only a representative for the conjugacy class. Here we derive a new iteration scheme that gives either the unique palindrome in the conjugacy class or expresses the word as a unique product of two unique palindromes. We denote these words by Ep/q where p/q is rational number expressed in lowest terms. We prove that Ep/q is a palindrome if pq is even and the unique product of two unique palindromes if pq is odd. We prove that the pairs (Ep/q, Er/s) generate the group when |ps − rq| = 1. This improves the previously known result that held only for pq and rs both even. The derivation of the enumeration scheme also gives a new proof of the known results about primitives.