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This paper extends and applies algebraic invariants and constructions formixing finite group extensions of shifts of finite type. For a finite abeliangroup G, Parry showed how to define a G-extension S_A from a square matrix Aover Z_+G, and classified the extensions up to topological conjugacy by thestrong shift equivalence class of A over Z_+G. Parry asked in this case if thedet(I-tA) (which captures the "periodic data" of the extension) would classifyup to finitely many topological conjugacy classes the extensions by G of afixed mixing shift of finite type. When the algebraic K-theory group NK_1(ZG)is nontrivial (e.g., for G=Z/4), we show the dynamical zeta function for anysuch extension is consistent with infinitely many topological conjugacyclasses. Independent of NK_1(ZG): for every nontrivial abelian G we show thereexists a shift of finite type with an infinite family of mixing nonconjugate Gextensions with the same dynamical zeta function. We define computable completeinvariants for the periodic data of the extension for G not necessarilyabelian, and extend all the above results to the nonabelian case. There isother work on basic invariants. The constructions require the "positiveK-theory" setting for positive equivalence of matrices over ZG[t].

Finite group extensions of shifts of finite type: K-theory, Parry and Liv\v{s}ic