Finite group extensions of shifts of finite type: K-theory, Parry and Livšic

This paper extends and applies algebraic invariants and constructions for mixing finite group extensions of shifts of finite type. For a finite abelian group G, Parry showed how to define a G-extension SA from a square matrix over ℤ₊G, and classified the extensions up to topological conjugacy by the strong shift equivalence class of A over ZCG. Parry asked, in this case, if the dynamical zeta function det(I - t A)^-1 (which captures the 'periodic data' of the extension) would classify the extensions by G of a fixed mixing shift of finite type up to a finite number of topological conjugacy classes. When the algebraic K-theory group NK₁(ℤG) is non-trivial (e.g. for G = ℤ/n with n not squarefree) and the mixing shift of finite type is not just a fixed point, we show that the dynamical zeta function for any such extension is consistent with an infinite number of topological conjugacy classes. Independent of NK₁(ℤG), for every non-trivial abelian G we show that there exists a shift of finite type with an infinite family of mixing non-conjugate G extensions with the same dynamical zeta function. We define computable complete invariants for the periodic data of the extension for G (not necessarily abelian), and extend all the above results to the non-abelian case. There is other work on basic invariants. The constructions require the 'positive K-theory' setting for positive equivalence of matrices over ℤG[t].