Restricted Permutations and Permanents of Infinite Amenable Groups

TL;DR

This paper explores the dynamical properties of certain shift spaces over infinite amenable groups, linking topological entropy to permanents of matrices and introducing a new notion of permanent for elements in the group ring, with conceptual connections to topological pressure.

Contribution

It introduces a novel approach to studying shift spaces over amenable groups by relating entropy to permanents and defining a new permanent concept in the group ring, offering new insights into their structure.

Findings

Topological entropy expressed as growth rate of permanents

Introduction of permanent for elements in the group ring

Conceptual link between permanent and topological pressure

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 $\textrm{h}_{\textrm{top}}(X_\mathsf{A})$ 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 $X_\mathsf{A}$.