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

Author:

Kapovich Ilya1,Simon Zachary2

Affiliation:

1. Department of Mathematics and Statistics, Hunter College of CUNY, 695 Park Ave, New York, NY 10065, USA

2. Advanced Technology Center, Lockheed Martin Space, 1111 Lockheed Martin Way, Sunnyvale, CA 94089, USA

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, [Formula: see text] and [Formula: see text], where [Formula: see text], and obtained some upper and lower bounds for these functions. In this paper, we study the behavior of the sequence [Formula: see text] as [Formula: see text]. Answering a question from [17], we prove that this sequence is unbounded and that for [Formula: see text], we have [Formula: see text]. By contrast, we show that for all [Formula: see text], one has [Formula: see text]. 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.

Funder

NSF

Publisher

World Scientific Pub Co Pte Ltd

Subject

General Mathematics

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