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 \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}(\lvert f\rvert) of the absolute value \lvert f\rvert 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}} .