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Local limit theorems for free groups. (English) Zbl 1012.20019

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.

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
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