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

TL;DR

This paper investigates algebraic invariants of finite group extensions of shifts of finite type, revealing that the dynamical zeta function does not always classify extensions up to conjugacy, especially when certain K-theory groups are nontrivial.

Contribution

It introduces new invariants for classifying group extensions of shifts of finite type, extending results to nonabelian groups and demonstrating limitations of zeta functions for classification.

Findings

Dynamical zeta functions can correspond to infinitely many conjugacy classes when NK_1(ZG) is nontrivial.

Existence of infinite families of nonconjugate extensions with identical zeta functions for nontrivial abelian groups.

Development of computable invariants for periodic data in nonabelian group extensions.

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 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].