Primitivity rank for random elements in free groups (Q2170266)

An element \(g\) of a free group \(F\) is \textit{primitive} in \(F\) if \(g\) belongs to some free basis of \(F\). The \textit{primitivity rank} \(\pi(g)\) of a non-trivial element \(g\) of the free group \(F_r\) of rank \(r\geq 2\) is a measure of the algebraic complexity of elements of \(F_r\) that was introduced by \textit{D. Puder} [Isr. J. Math. 201, Part A, 25--73 (2014; Zbl 1308.20023)]. It is defined to be the smallest rank of a subgroup \(H\) of \(F\) containing \(g\) as a non-primitive element, if such a subgroup \(H\) exists, and \(\pi(g)=\infty\) otherwise. The \textit{critical set} \(\mathrm{Crit}(g)\) of a non-trivial element \(g\in F_r\) is the set of all subgroups \(H\) of \(F_r\) containing \(g\) as a non-primitive element and such that \(\mathrm{rank}(H)=\pi(g)\). The main result of this paper is that there is an exponentially generic subset \(Y_r \subset F_r\) (in the sense of \textit{I. Kapovich} et al. [J. Algebra 264, No. 2, 665--694 (2003; Zbl 1041.20021)]) such that for every \(w\in Y_r\) we have \(\pi(w)=r\) (the largest possible) and \(\mathrm{Crit}(w)=\{F_r\}\). The analogous result holds if \(F_r\) is replaced by \(U_r\), the set of all cyclically reduced words in \(F_r\). Moreover, in the case of \(F_r\), the element \(w\) can be produced by a simple non-backtracking random walk of length \(n\) in \(F_r\). Combining this result with a result of \textit{D. Puder} and \textit{O. Parzanchevski} [J. Am. Math. Soc. 28, No. 1, 63--97 (2015; Zbl 1402.20042)] provides precise asymptotics for the number of fixed points of a random permutation \(\sigma\) of the symmetric group \(S_N\) with respect to the word measure on \(S_N\) defined by a generic word \(w \in F_r\). The proof employs genericity conditions of \textit{G. N. Arzhantseva} [Fundam. Prikl. Mat. 3, No. 3, 675--683 (1997; Zbl 0929.20025)], \textit{G. N. Arzhantseva} and \textit{A. Yu. Ol'shanskij} [Math. Notes 59, No. 4, 350--355 (1996; Zbl 0877.20021); translation from Mat. Zametki 59, No. 4, 489--496 (1996)] applied to Stallings subgroup graphs \(\Gamma_H\) of subgroups \(H\) of \(F_r\).