Growth of primitive elements in free groups. (Q2874650)

Let \(F_k\) be the free group on \(k\geq 2\) generators. An element \(w\in F_k\) is primitive if it belongs to some basis of \(F_k\); a subgroup \(H\leq F_k\) is a free factor of \(F_k\) if there is another group \(H'\) such that \(H*H'=F_k\). Let \(P_{k,N}=\{w\in F_k\mid w\mathrm{ is primitive},\;|w|=N\}\), \(C_{k,N}=\{[w]\in F_k\mid w\mathrm{ is primitive},\;|w|_c=N\}\), where \([w]\), the cyclic word associated with \(w\), denotes the conjugacy class of \(w\); \(|w|\) denotes the length of \(w\) in \(F_k\), and \(|w|_c\), the cyclic length of \(w\), denotes the length of cyclically reduced elements of \([w]\).NEWLINENEWLINE Wicks asked for the growth rates \(\rho_k=\limsup_{N\to\infty}(|P_{k,N}|^{1/N})\) and \(\limsup_{N\to\infty}(|C_{k,N}|^{1/N})\). This article answers these questions: for \(k\geq 3\) \(\rho_k=2k-3\) (\(\rho_2=\sqrt 3\) is known by \textit{I. Rivin} [Geom. Dedicata 107, 99-100 (2004; Zbl 1071.20032)]), and for \(k\geq 2\) we have \(\limsup_{N\to\infty}(|C_{k,N}|^{1/N})=2k-3\). The proofs involve a detailed analysis of the Whitehead algorithm to detect primitive elements and understanding random primitive words of length \(N\). It is shown that such a word contains one of the generators of \(F_k\) exactly once asymptotically almost surely (as \(N\to\infty\)). The method of proof is also used to find the growth rate of the set \(S_{k,N}\) (containing \(P_{k,N}\)) that consists of all elements of \(F_k\) of length \(N\) that are contained in a proper free factor of \(F_k\), and for \(k\geq 3\) we have \(\limsup_{N\to\infty}(|S_{k,N}|^{1/N})=2k-3\).