Extensions of Cantor Minimal Systems and Dimension Groups

Given a factor map p : (X, T )→ (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 K0(X)/K0(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(X | 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(X | Y ) are torsion groups. As a consequence we can now identify H(X | Y ) as the torsion group of K0(X)/K0(Y ).