Restricted permutations and permanents of infinite amenable groups

Abstract

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 ${\mathrm{h}}_{\mathrm{top}}(X_{\mathsf{A}})$ can be expressed as logarithmic growth rate of permanents of certain finite $(0,1)$-matrices associated with right Følner sequences in $\Gamma$. Motivated by the difficulty of computing such permanents, we introduce the notion of the permanent $\mathsf{Per}(f)$ for nonnegative elements $f$ in the real group ring $\mathbb{R}\Gamma$ of $\Gamma$ whose support is the alphabet $\mathsf{A}$ of the SFT $X_{\mathsf{A}}$, and compare, for arbitrary $f\in \mathbb{R}\Gamma$, the Fuglede–Kadison determinant ${\mathsf{Det}}_{\mathrm{FK}}(f)$ with the permanent $\mathsf{Per}(|f|)$ 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 logarithmic permanent $\mathsf{per}(f)=\log \mathsf{Per}(f)$ of a nonnegative element $f\in \mathbb{R}\Gamma$ can be viewed as topological pressure of the restricted-permutation SFT $X_{\mathsf{A}}$ associated with the function $\log f$ on the alphabet $\mathsf{A}=\mathrm{supp}(f)$ of $X_{\mathsf{A}}$.