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

TL;DR

This paper investigates the primitivity index in free groups, demonstrating its unbounded growth along certain sequences and contrasting it with the bounded simplicity index, using topological, group-theoretic, and number-theoretic methods.

Contribution

It proves the unboundedness of the primitivity index sequence and its asymptotic relation to the second Chebyshev function, providing new insights into free group element complexity.

Findings

d_{prim}(a^nb^n; F(a,b)) is unbounded as n→∞

d_{simp}(a^nb^n; F(a,b))=2 for all n≥2

Sequence |d_{prim}(a^{n_i}b^{n_i}) - log(n_i)| ≤ o(log(n_i)) for n_i=lcm(1,2,...,i)

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, $d_{prim}(g;F_N)$ and $d_{simp}(g;F_N)$, where $1\ne g\in F_N$, and obtained some upper and lower bounds for these functions. In this paper, we study the behavior of the sequence $d_{prim}(a^nb^n; F(a,b))$ as $n\to\infty$. Answering a question of Kapovich, we prove that this sequence is unbounded and that for $n_i=lcm(1,2,\dots,i)$, we have $|d_{prim}(a^{n_i}b^{n_i}; F(a,b))-\log(n_i)|\le o(\log(n_i))$. By contrast, we show that for all $n\ge 2$, one has $d_{simp}(a^nb^n; F(a,b))=2$. 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.