Primitivity rank for random elements in free groups

Abstract

For a free group Fr of finite rank r ≥ 2 and a nontrivial element w ∈ Fr the primitivity rank π(w) is the smallest rank of a subgroup H ≤ Fr such that w ∈ H and that w is not primitive in H (if no such H exists, one puts π(w) = ∞). The set of all subgroups of Fr of rank π(w) containing w as a non-primitive element is denoted Crit(w). These notions were introduced by Puder in [20]. We prove that there exists an exponentially generic subset V ⊆ Fr such that for every w ∈ V we have π(w) = r and Crit(w) = {Fr}.