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Kapovich, Ilya Schupp, Paul Leininger, Chris Mineyev, Igor Gupta, Neha 2016-11-10T17:49:21Z 2016-11-10T17:49:21Z 2016-05-03 2016-08 This thesis primarily addresses the problem of untangling closed geodesics in finite covers of hyperbolic surfaces. Our motivation comes from results of Scott and Patel. Scott's result tells us that one can always untangle a closed geodesic on a hyperbolic surface in a finite degree cover. Our goal is to quantify the degree of this cover in which the geodesic untangles in terms of the length of the geodesic. Our approach is to introduce and study the notions of primitivity, simplicity and non-filling index functions for finitely generated free groups. In joint work with Ilya Kapovich we obtain lower bounds for these functions and relate these free group results back to the setting of hyperbolic surfaces. Chapters 1-6 in parts comprise of a joint paper with Kapovich that is under review. Chapter 7 discusses the problem of Nielsen equivalence in a particular class of generic groups. Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2016-11-09 without embargo terms The student, Neha Gupta, accepted the attached license on 2016-05-02 at 11:33. The student, Neha Gupta, submitted this Dissertation for approval on 2016-05-02 at 11:40. This Dissertation was approved for publication on 2016-05-03 at 08:39. DSpace SAF Submission Ingestion Package generated from Vireo submission #9594 on 2016-11-09 at 10:19:18 Made available in DSpace on 2016-11-10T17:49:21Z (GMT). No. of bitstreams: 3 GUPTA-DISSERTATION-2016.pdf: 791379 bytes, checksum: 18baa8d997b404bc08de2a2fda20863b (MD5) LICENSE.txt: 4207 bytes, checksum: 7d7919b4e828a0ee69b1b9baf3de7f96 (MD5) PROQUEST_LICENSE.txt: 4553 bytes, checksum: 2c72a85f3b002ca3e0d6e7f3568eef69 (MD5) Previous issue date: 2016-05-03 application/pdf http://hdl.handle.net/2142/92691 en Copyright 2016 Neha Gupta Untangling curves free groups Certain free group functions and untangling closed curves on surfaces text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.