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 Z+G. Parry asked in this case if the dynamical zeta function det(I − tA)−1 (which captures the “periodic data” of the extension) would classify up to finitely many topological conjugacy classes the extensions by G of a fixed mixing shift of finite type. When the algebraic K-theory group NK1(ZG) is nontrivial (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 the dynamical zeta function for any such extension is consistent with infinitely many topological conjugacy classes. Independent of NK1(ZG), for every nontrivial abelian G we show there exists a shift of finite type with an infinite family of mixing nonconjugate 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 nonabelian case. There is other work on basic invariants. The constructions require the “positive K-theory” setting for positive equivalence of matrices over ZG[t].