Densities in free groups and ℤk, visible points and test elements

Ilya Kapovich, Igor Rivin, Paul Schupp, Vladimir Shpilrain

Research output: Contribution to journalArticlepeer-review


In this article we relate two different densities. Let Fk be the free group of finite rank k > 2 and let a 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 1/tkζ(k), where ζis 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.

Original languageEnglish (US)
Pages (from-to)263-284
Number of pages22
JournalMathematical Research Letters
Issue number2-3
StatePublished - 2007

ASJC Scopus subject areas

  • Mathematics(all)


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