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Finite group extensions of shifts of finite type: $K$-theory, Parry and Livšic
Published online by Cambridge University Press: 11 February 2016
Abstract
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 $S_{A}$ from a square matrix over $\mathbb{Z}_{+}G$, and classified the extensions up to topological conjugacy by the strong shift equivalence class of $A$ over $\mathbb{Z}_{+}G$. Parry asked, in this case, if the dynamical zeta function $\det (I-tA)^{-1}$ (which captures the ‘periodic data’ of the extension) would classify the extensions by $G$ of a fixed mixing shift of finite type up to a finite number of topological conjugacy classes. When the algebraic $\text{K}$-theory group $\text{NK}_{1}(\mathbb{Z}G)$ is non-trivial (e.g. for $G=\mathbb{Z}/n$ with $n$ not square-free) 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 $\text{NK}_{1}(\mathbb{Z}G)$, 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 $\mathbb{Z}G[t]$.
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- © Cambridge University Press, 2016
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