Introduction

In this paper we discuss several more or less well-known theorems about primitive and palindromic words in two generator free groups. We describe a geometric technique that ties all of these theorems together and gives new proofs of all but the last of them, which is an enumerative scheme for palindromic words. This geometric approach and the enumerative scheme will be useful in applications. These applications will be studied elsewhere [GK3].

The main object here is a two generator free group which we denote by G = 〈A, B〉.

Definition 1.1A word W = W(A, B) ∈ G is primitive if there is another word V = V (A, B) ∈ G such that W and V generate G. V is called a primitive associate of W and the unordered pair W and V is called a pair of primitive associates.

We remark that if W, V is a pair of primitive associates then both WV and WV−1 are primitive and W, WV±1 and V, WV±1 are both primitive pairs.

Definition 1.2Aword W = W (A, B) ∈ G is a palindrome if it reads the same forward and backward.

In [GK1] we found connections between a number of dfferent forms of primitive words and pairs of primitive associates in a two generator free group. These were obtained using both algebra and geometry. The theorems that we discuss, Theorems 2.9, 2.10 and 2.11, can be found in [GK1] and Theorem 2.13 can be found in Piggott [P].