Primitivity index bounds in free groups, and the second Chebyshev function

Ilya Kapovich; Zachary Simon

paper

Primitivity index bounds in free groups, and the second Chebyshev function

arXiv:2012.13655

Abstract

Motivated by results about "untangling" closed curves on hyperbolic surfaces, Gupta and Kapovich introduced the primitivity and simplicity index functions for finitely generated free groups, $d_{p r im} (g; F_{N})$ and $d_{s im p} (g; F_{N})$ , where $1 \neq = g \in F_{N}$ , and obtained some upper and lower bounds for these functions. In this paper, we study the behavior of the sequence $d_{p r im} (a^{n} b^{n}; F (a, b))$ as $n \to \infty$ . Answering a question of Kapovich, we prove that this sequence is unbounded and that for $n_{i} = l c m (1, 2, \dots, i)$ , we have $∣ d_{p r im} (a^{n_{i}} b^{n_{i}}; F (a, b)) - lo g (n_{i}) ∣ \leq o (lo g (n_{i}))$ . By contrast, we show that for all $n \geq 2$ , one has $d_{s im p} (a^{n} b^{n}; F (a, b)) = 2$ . In addition to topological and group-theoretic arguments, number-theoretic considerations, particularly the use of asymptotic properties of the second Chebyshev function, turn out to play a key role in the proofs.

16 pages, 3 figures; to appear in International Journal of Algebra and Computation