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

Published online by Cambridge University Press:  17 October 2017

NEHA GUPTA  and
ILYA KAPOVICH
NEHA GUPTA
Affiliation:
Department of Mathematics, FAS, Harvard University, One Oxford Street, Cambridge, MA 02138, U.S.A. e-mail: neha@math.harvard.edu
ILYA KAPOVICH
Affiliation:
Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 West Green Street, Urbana, IL 61801, U.S.A. e-mail: kapovich@math.uiuc.edu
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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 fprim(n, FN) to the residual finiteness growth function for FN.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 1 , January 2019 , pp. 83 - 121
Copyright
Copyright © Cambridge Philosophical Society 2017 

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