The primitivity index function for a free group, and untangling closed curves on hyperbolic surfaces.With the 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 f prim (n, F N ) to the residual finiteness growth function for F N .

Decision problems, complexity, traces, and representations

“…In particular, the degree of the polynomial in Corollary 1.7 depends only on m and the coefficient ring of the representation, both of which are constant for a fixed γ. By Patel [51] and Gupta-Kapovich [28], we have m ||γ||, and so when ||γ|| , ||η|| ≤ n, we see that CD Γ (γ, η) ≤ Cn Cn 2 .…”

“…In the case of surface groups, it is directly related to the smallest degree of a cover where the curve associated to γ has a lift to a simple closed curve. Patel [51] and Gupta-Kapovich [28] have given upper bounds in the case of surface groups and free groups, respectively, on order ||γ||. Gaster [24], improving on work of Gupta-Kapovich [28], has shown that there exist γ that require a cover of degree on the order of ||γ||.…”

Building hyperbolic metrics suited to closed curves and applications to lifting simply

“…In particular, for ρ 0 one has f ρ0 (L) < 16.2 · L [14, Thm 1.1]. Gupta-Kapovich then gave a lower bound on f ρ (L) [7] that was recently improved upon by the second author. Combining the results of the second author [6] and third author [14] gives the following description of f ρ (L) found in [6, pp.…”

“…For a closed curve γ on S, let deg(γ) be the smallest degree of a cover where γ lifts simply. Recently, there has been an effort to prove effective versions of Scott's result by obtaining bounds on deg(γ) [6,7,14]. One of the main goals of this paper is to study the dependence of deg(γ) on two other notions of complexity; the geometric self-intersection number i(γ, γ) of the curve, and, fixing a complete hyperbolic metric ρ on S, the length ρ (γ).…”

Lifting Curves Simply

“…The 'bigon criterion' of [FM,§1.2.4] can be altered straightforwardly to an 'immersed bigon criterion' in the setting of curves with self-intersections, and so the lack of immersed bigons guarantees that the chosen representative of γ n is in minimal position. We use Lemma 5 to estimate deg(γ n ), a calculation reminiscent of [GK,Lemma 3.10]. The following proposition is the main tool in our analysis.…”