有限型移位的有限群扩张：K-理论，Parry和Livšic
Finite group extensions of shifts of finite type: K-theory, Parry and Livšic
摘要：
本文推广和应用了有限型移位的混合有限群扩张的代数不变式和构造。对于有限阿贝尔群G，Parry给出了如何定义Z +G上方阵A的G-扩张SA，并利用A在Z +G上的强移位等价类将扩张分类到拓扑共轭.帕里问在这种情况下，如果det（I-tA）（它捕获的“周期性数据”的扩展）将分类多达1000拓扑共轭类的扩展由G的固定混合移位的有限型。当代数K理论群NK_1（ZG）是非平凡的（例如，对于G=Z/4），我们证明了任意扩张的动态zeta函数与无穷多个拓扑共轭类是一致的.与NK_1（ZG）无关：对每个非平凡的阿贝尔G，我们证明了存在一个有限型移位，它具有一个无限族的混合非共轭G扩张，这些扩张具有相同的动态zeta函数。我们定义了G的扩张的周期数据的可计算完全不变量，并将上述结果推广到非阿贝尔情形。还有其他关于基本不变式的工作。这些构造需要ZG[t]上矩阵的正等价的“正K-理论”设置。
英文摘要：
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 A 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 det(I-tA) (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 NK_1(ZG) is nontrivial (e.g., for G=Z/4), we show the dynamical zeta function for any such extension is consistent with infinitely many topological conjugacy classes. Independent of NK_1(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].
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