Counting Primitive Elements in Free Groups

In this paper it is proved that the set of primitive elements of a nonabelian free group has density zero, i.e. the ratio of primitive elements in increasingly large balls is arbitrarily small. Two notions of density (natural and exponential density) are defined and some of their properties are studied. A class of subsets of the free group (graphical sets) is defined restricting the occurrence of adjacent letters in the reduced word for an element, and the relation between graphical sets and the set of primitive elements is studied and used to prove the above result. Primitive elements in free groups are those that can be part of a free basis of the group. There has been a large body of research concerning primitive elements, dating back to the original paper by Whitehead [W]. It is well known, for instance, what a primitive element looks like in the free group of rank 2 (Cohen et. al., [CMZ] and Hoare, [H]). In this paper we will be interested in their distribution inside the free group with relation to the concentric balls of elements. In particular, we will prove that the set of primitive elements is increasingly sparse in subsequent balls, or, using our terminology, that its natural density is zero, and we will give more precise bounds of their number in each ball, using the exponential density.