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**1 - 1**of**1**### THE PRIMITIVITY INDEX FUNCTION FOR A FREE GROUP, AND UNTANGLING CLOSED CURVES ON HYPERBOLIC SURFACES

"... Abstract. Scott [39] proved that if Σ is a closed surface with a hyperbolic metric ρ, then for every closed geodesic γ on Σ there exists a finite cover of Σ where γ lifts to a simple closed geodesic. Define fρ(L) ≥ 0 to be the smallest monotone nondecreasing function such that every closed geodesic ..."

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Abstract. Scott [39] proved that if Σ is a closed surface with a hyperbolic metric ρ, then for every closed geodesic γ on Σ there exists a finite cover of Σ where γ lifts to a simple closed geodesic. Define fρ(L) ≥ 0 to be the smallest monotone nondecreasing function such that every closed geodesic of length ≤ L on Σ lifts to a simple closed geodesic in a cover of Σ of degree ≤ fρ(L). A result of Patel [32] implies that for every hyperbolic structure ρ on Σ there exists K = K(ρ)> 0 such that fρ(L) ≤ KL for all L> 0. We prove that there exist c = c(ρ)> 0 such that fρ(L) ≥ c log1/3 L for all sufficiently large L. We obtain a similar lower bound for the function fΣ defined analogously to fρ but using the self-intersection number of closed curves on Σ instead of the hyperbolic length. These results are obtained as a consequence of several related results that we establish for free groups. Thus we define, study and obtain lower bounds for the primitivity index function f(n) and the simplicity index function f0(n) for the free group FN = F (a1,..., aN) of finite rank N ≥ 2. The primitivity index function f(n) is the smallest monotone non-decreasing function f(n) ≥ 0 such that for every nontrivial freely reduced word w ∈ FN of length ≤ n there is a subgroup H ≤ FN of index ≤ f(n) such that w ∈ H and that w is a primitive element (i.e. an element of a free basis) of H. The function f0(n) is defined similarly except that instead of w being primitive in H we require that w belongs to a proper free factor of H. The lower bounds for f(n) and f0(n) are obtained via probabilistic methods, by estimating from below the simplicity index for a “sufficiently random ” element wn ∈ FN produced by a simple non-backtracking random walk of length n on FN. 1.