Densities in free groups and Zᵏ, Visible Points and Test Elements

Original · EN

In this article we relate two different densities. Let Fₖ be the free group of finite rank k ≥ 2 and let α be the abelianization map from Fₖ onto Zᵏ. We prove that if S Zᵏ is invariant under the natural action of SL(k, Z) then the asymptotic density of S in Zᵏ and the annular density of its full preimage α⁻¹(S) in Fₖ are equal. This implies, in particular, that for every integer t≥ 1, the annular density of the set of elements in Fₖ that map to t-th powers of primitive elements in Zᵏ 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 annular density of the set of all test elements in the free group F(a,b) of rank two is 1-6/π². Equivalently, this shows that the union of all proper retracts in F(a,b) has annular density 6/π². 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.