Restricted permutations and permanents of infinite amenable groups

Let Γ be an infinite discrete group and A⊂ Γ a nonempty finite subset. The set of permutations σ of Γ such that s⁻¹σ (s)∈ A for every s∈ Γ can be identified with a shift of finite type XA⊂ AΓ over Γ . In this paper, we study dynamical properties of such shift spaces, like invariant probability measures, topological entropy, and topological pressure, under the hypothesis that Γ is amenable. In this case, the topological entropy hₜₒₚ(XA) can be expressed as logarithmic growth rate of permanents of certain finite (0,1) -matrices associated with right Følner sequences in Γ . Motivated by the difficulty of computing such permanents, we introduce the notion of the permanent Per(f) for nonnegative elements f in the real group ring RΓ of Γ whose support is the alphabet A of the SFT XA , and compare, for arbitrary f ∈ RΓ , the Fuglede–Kadison determinant DetFK(f) with the permanent 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 per(f) = log Per(f) of a nonnegative element f∈ RΓ can be viewed as topological pressure of the restricted-permutation SFT XA associated with the function log f on the alphabet A=supp(f) of XA .