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.
Categories: math.GR math.GT math.NT