2005/07/27 by Ilya Kapovich, Kapovich, Ilya, Igor Rivin +6
Computer Science · Mathematics · #20F #37A #60B #60F #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Primary 20P05 #Secondary 11M #Topological and Geometric Data Analysis #math.GR #math.GT #msc:11M #msc:20F #msc:20P05 #msc:37A #msc:60B #msc:60F
paper · pdf · doi:10.48550/arxiv.math/0507573
Revised and corrected version, reflecting the correct statement of the local limit theorem
openalex publication_date 2005/07/27 · arxiv created 2005/11/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathbb 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.