Extensions of Cantor minimal systems and dimension groups

TL;DR

This paper investigates the relationship between dimension groups of Cantor minimal systems connected by factor maps, exploring how intermediate extensions influence these groups and introducing higher order cohomology groups with torsion properties.

Contribution

It provides a new interpretation of torsion in dimension groups via intermediate extensions and defines higher order cohomology groups for Cantor minimal systems.

Findings

Torsion subgroup of $K_0(X)/K_0(Y)$ relates to intermediate abelian extensions.

Existence of non-abelian finite group extensions can embed $K_0(Y)$ into a proper subgroup of $K_0(X)$.

All higher order cohomology groups $H^n(X igm| Y)$ are torsion groups.

Abstract

Given a factor map $p : (X,T) \to (Y,S)$ of Cantor minimal systems, we study the relations between the dimension groups of the two systems. First, we interpret the torsion subgroup of the quotient of the dimension groups $K_0(X)/K_0(Y)$ in terms of intermediate extensions which are extensions of $(Y,S)$ by a compact abelian group. Then we show that, by contrast, the existence of an intermediate non-abelian finite group extension can produce a situation where the dimension group of $(Y,S)$ embeds into a proper subgroup of the dimension group of $(X,T)$, yet the quotient of the dimension groups is nonetheless torsion free. Next we define higher order cohomology groups $H^n(X \mid Y)$ associated to an extension, and study them in various cases (proximal extensions, extensions by, not necessarily abelian, finite groups, etc.). Our main result here is that all the cohomology groups $H^n(X \mid Y)$ are torsion groups. As a consequence we can now identify $H^0(X \mid Y)$ as the torsion group of $ K_0(X)/K_0(Y)$.