### Abstract

In this article we relate two different densities. Let F_{k} be the free group of finite rank k > 2 and let a be the abelianization map from F_{k} onto ℤ^{k}. We prove that if S ⊆ ℤ^{k} is invariant under the natural action of SL(k, ℤ) then the asymptotic density of S in ℤ^{k} and the annular density of its full preimage α^{-1}(S) in F_{k} are equal. This implies, in particular, that for every integer t > 1, the annular density of the set of elements in F_{k} that map to t-th powers of primitive elements in ℤ^{k} is equal to 1/t^{k}ζ(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/π^{2}. Equivalently, this shows that the union of all proper retracts in F(a, b) has annular density 6/π^{2}. 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 language | English (US) |
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Pages (from-to) | 263-284 |

Number of pages | 22 |

Journal | Mathematical Research Letters |

Volume | 14 |

Issue number | 2-3 |

DOIs | |

State | Published - Jan 1 2007 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

^{k}, visible points and test elements.

*Mathematical Research Letters*,

*14*(2-3), 263-284. https://doi.org/10.4310/MRL.2007.v14.n2.a9