Sharp, Richard Local limit theorems for free groups. (English) Zbl 1012.20019 Math. Ann. 321, No. 4, 889-904 (2001). Let \(G\) be the free group on \(k\geq 2\) generators. For \(g\in G\), let \(|g|\) be its word length and \([g]\in\mathbb{Z}^k\) its image under the Abelianization map \(G\to\mathbb{Z}^k\). Let \(W(n)=\{g\in G:|g|=n\}\) and, for fixed \(\alpha=(\alpha_1,\dots,\alpha_k)\in\mathbb{Z}^k\), let \(W(n,\alpha)=\{g\in W(n):[g]=\alpha\}\). This paper is about the distribution of the elements of \(W(n)\) over \(\mathbb{Z}^k\) and the dependence of \(|W(n,\alpha)|\) on \(\alpha\) as well as on \(n\). The approach is to regard \(|W(n,\alpha)|/|W(n)|\) as a probability distribution on \(\mathbb{Z}^k\) and to study the asymptotic behavior as \(n\to\infty\). Since one of \(|W(n,\alpha)|\) or \(|W(n+1,\alpha)|\) is zero depending on the parity of \(\alpha_1+\cdots+\alpha_k\), it is reasonable to consider the quantity \(Q(n,\alpha)=|W(n,\alpha)|/|W(n)|+|W(n+1,\alpha)|/|W(n+1)|\). The main result is a local limit theorem: \[ \lim_{n\to\infty}\left|\sigma^kn^{k/2}Q(n,\alpha)-\frac 2{(2\pi)^{k/2}}e^{-\|\alpha\|^2/2\sigma^2n}\right|=0, \] uniformly in \(\alpha\) where \(\sigma^2\) is an appropriate variance depending on \(k\). The theorem is obtained by using subshifts of finite type and proving a similar theorem for certain maps \(X_A\to\mathbb{Z}^k\) (where \(X_A\) is the shift space) playing the role of the Abelianization map. A related result for fundamental groups of compact surfaces of genus \(\geq 2\) is also obtained. Reviewer: Bruce Hughes (Nashville) Cited in 14 Documents MSC: 20E05 Free nonabelian groups 37B10 Symbolic dynamics 20F65 Geometric group theory 20F67 Hyperbolic groups and nonpositively curved groups 20P05 Probabilistic methods in group theory 37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems 57M05 Fundamental group, presentations, free differential calculus Keywords:free groups; subshifts of finite type; strong Markov property; hyperbolic groups; probability distributions; fundamental groups PDFBibTeX XMLCite \textit{R. Sharp}, Math. Ann. 321, No. 4, 889--904 (2001; Zbl 1012.20019) Full Text: DOI