Restricted Permutations and Permanents of Infinite Amenable Groups

Let $\Gamma$ be an infinite discrete group and $\mathsf{A}\subset \Gamma$ a nonempty finite subset. The set of permutations $\sigma$ of $\Gamma$ such that $s^{-1}\sigma (s)\in \mathsf{A}$ for every $s\in \Gamma$ can be identified with a shift of finite type $X_\mathsf{A}\subset \mathsf{A}^{\Gamma}$ over $\Gamma$. In this paper we study dynamical properties of such shift spaces, like invariant probability measures, topological entropy, and topological pressure, under the hypothesis that $\Gamma$ is amenable. In this case the topological entropy $\textrm{h}_{\textrm{top}}(X_\mathsf{A})$ can be expressed as logarithmic growth rate of permanents of certain finite (0,1)-matrices associated with right F{\o}lner sequences in $\Gamma$. Motivated by the difficulty of computing such permanents we introduce the notion of the permanent $\textrm{per}(f)$ for nonnegative elements $f$ in the real group ring $\mathbb{R}\Gamma$ of $\Gamma$ whose support is the alphabet $\mathsf{A}$ of the shift space $X_\mathsf{A}$, and compare, for arbitrary $f \in \mathbb{R}\Gamma$, the Fuglede-Kadison determinant $\textrm{det} _\textrm{FK}(f)$ with the permanent $\textrm{per}(|f|)$ of the absolute value $|f|$ of $f$. Although this approach is effective in only few examples, discussed below, it is interesting from a conceptual point of view that the permanent $\textrm{per}(f)$ of a nonnegative element $f\in \mathbb{R}\Gamma$ can be viewed as topological pressure of the restricted-permutation shift space $X_\mathsf{A}$ associated with the function $\log f$ on the alphabet $\mathsf{A}=\textrm{supp}(f)$ of $X_\mathsf{A}$.