Asymptotic Density in Free Groups and Z, Visibile Points and Test Elements

Let Fk be the free group of finite rank k ≥ 2 and let α be the abelianization map from Fk onto Z . We prove that if S ⊆ Z is invariant under the natural action of SL(k,Z) then the asymptotic density of S in Z and the asymptotic density of its full preimage α−1(S) in Fk are equal. This implies, in particular, that for every integer t ≥ 1, the asymptotic density of the set of elements in Fk that map to t-th powers of primitive elements in Z k is equal to to 1 tkζ(k) , where ζ is the Riemann zeta-function. An element g of a group G is called a test element if every endomorphism of G which fixes g is an automorphism of G. As an application of the result above we prove that the asymptotic density of the set of all test elements in the free group F (a, b) of rank two is 1 − 6 π . (Equivalently, this shows that the union of all proper retracts in F (a, b) has asymptotic density 6 π .) Thus being a test element in F (a, b) is an “intermediate property” in the sense that the probability of being a test element is strictly between 0 and 1.