Ilya Kapovich: Primitivity rank for random elements in free groups

In 2014 Doron Puder introduced the notion of primitivity rank π(g) for a non-trivial element g in a free group F_r of rank r.

Namely, π(g) is defined as the smallest rank of a subgroup H of F_r containing g as a non-primitive element, or as ∞ if no such H exists. The set of all subgroups H of F_r as above is denoted Crit(g). It turns out that the primitivity index of an element w ∈ F_r is closely related to the questions about word-hyperbolicity and subgroup properties of the one-relator group < F_r | w=1 >.

We prove that if r≥2 and F₂=F(x₁, …, x_r) is the free group of rank r, then, as n→∞, for a “random” element w_n ∈ F_r of length n with probability tending to 1 one has π(w)=r and Crit(w)={F_r}. We discuss applications of this result to “word measures” on finite symmetric groups S_N, defined by such w_n.

This video is part of the New York Group Theory Cooperative‘s group theory seminar series.