The primitivity index function for a free group, and untangling closed curves on hyperbolic surfaces. With the appendix by Khalid Bou-Rabee

Motivated by the results of Scott and Patel about "untangling" closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back to the setting of hyperbolic surfaces. An appendix by Khalid Bou-Rabee connects the primitivity index function f _prim (n, F _N ) to the residual finiteness growth function for F _N .