Certain free group functions and untangling closed curves on surfaces

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