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, dprim(g;FN) and dsimp(g;FN), where 1 ≠ g ∈ FN, and obtained some upper and lower bounds for these functions. In this talk, we study the behaviour of the sequence dprim(anbn; F(a,b)) as n → ∞. Answering a question of Kapovich, we prove that this sequence is unbounded and that for ni=lcm(1,2,…,i), we have |dprim(anibni; F(a,b))-log(ni)| = o(log(ni)). By contrast, we show that for all n ≥ 2, one has dsimp(anbn;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.
The talk is based on a joint paper with Zachary Simon.
This video is part of the New York Group Theory Cooperative‘s group theory seminar series.
