Primitivity rank for random elements in free groups

Abstract

Abstract For a free group F r F_{r} of finite rank r ≥ 2 r\geq 2 and a non-trivial element w ∈ F r w\in F_{r} , the primitivity rank π ⁢ ( w ) \pi(w) is the smallest rank of a subgroup H ≤ F r H\leq F_{r} such that w ∈ H w\in H and 𝑤 is not primitive in 𝐻 (if no such 𝐻 exists, one puts π ⁢ ( w ) = ∞ \pi(w)=\infty ). The set of all subgroups of F r F_{r} of rank π ⁢ ( w ) \pi(w) containing 𝑤 as a non-primitive element is denoted by Crit ⁡ ( w ) \operatorname{Crit}(w) . These notions were introduced by Puder (2014). We prove that there exists an exponentially generic subset V ⊆ F r V\subseteq F_{r} such that, for every w ∈ V w\in V , we have π ⁢ ( w ) = r \pi(w)=r and Crit ⁡ ( w ) = { F r } \operatorname{Crit}(w)=\{F_{r}\} .