Matsumoto K-groups associated to certain shift spaces

• Author: Toke M. Carlsen and Søren Eilers.
• Date: April 2004, final version December 2004.
• Status: Appeared in Documenta Mathematica 9 (2004), 639-671.
• Pages: 32
• Abstract: Matsumoto associated to each shift space (also called a subshift) an abelian group which is now known as Matsumoto's K₀-group. It is defined as the cokernel of a certain map and resembles the first cohomology group of the dynamical system which has been previously studied.

  In this paper, we will for shift spaces having a certain property (*), show that the first cohomology group is a factor group of Matsumoto's K₀-group. We will also for shift spaces having an additional property (**), describe Matsumoto's K₀-group in terms of the first cohomology group and some extra information determined by the left special elements of the shift space. We determine for a broad range of different classes of shift spaces if they have property (*) and property (**) and use this to show that Matsumoto's K0-group and the first cohomology group are isomorphic for example for finite shift spaces and for Sturmian shift spaces. Furthermore, the ground is laid for a description of the Matsumoto K₀-group as an ordered group in a forthcoming paper.

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