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

Abstract

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 to the residual finiteness growth function for $F_N$.

Final version. To appear in Math. Proc. Cambridge Phil. Soc

References in corpus (4)

• Hyperbolic extensions of free groups
• A train track directed random walk on $Out(F_r)$
• Characterizing linear groups in terms of growth properties
• Full residual finiteness growths of nilpotent groups

Cited by in corpus (2)

• The topology of local commensurability graphs
• Local commensurability graphs of solvable groups