Ring and module structures on dimension groups associated with a shift of finite type

TL;DR

This paper investigates algebraic invariants derived from K-theory of C*-algebras associated with shifts of finite type, providing explicit descriptions of their ring and module structures through inductive limits.

Contribution

It offers explicit descriptions of ring and module structures on K-groups for shifts of finite type, extending previous work on Smale spaces.

Findings

Explicit descriptions of K-theoretic invariants as inductive limits.

Ring and module structures on K-groups are characterized.

Connections to hyperbolic dynamics and Smale spaces are clarified.

Abstract

We study invariants for shifts of finite type obtained as the K-theory of various C*-algebras associated with them. These invariants have been studied intensely over the past thirty years since their introduction by Wolfgang Krieger. They may be given quite concrete descriptions as inductive limits of simplicially ordered free abelian groups. Shifts of finite type are special cases of Smale spaces and, in earlier work, the second author has shown that the hyperbolic structure of the dynamics in a Smale space induces natural ring and module structures on certain of these K-groups. Here, we restrict our attention to the special case of shifts of finite type and obtain explicit descriptions in terms of the inductive limits.