Z-actions on AH algebras and Z^2-actions on AF algebras

We consider Z-actions (single automorphisms) on a unital simple AH algebra with real rank zero and slow dimension growth and show that the uniform outerness implies the Rohlin property under some technical assumptions. Moreover, two Z-actions with the Rohlin property on such a C^*-algebra are shown to be cocycle conjugate if they are asymptotically unitarily equivalent. We also prove that locally approximately inner and uniformly outer Z^2-actions on a unital simple AF algebra with a unique trace have the Rohlin property and classify them up to cocycle conjugacy employing the OrderExt group as classification invariants.Comment: 24 page

Homology and topological full groups of etale groupoids on totally disconnected spaces

For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H_0 if and only if there exists an element in the topological full group which maps one to the other. It is also shown that a natural homomorphism, called the index map, from the topological full group to H_1 is surjective and any element of the kernel can be written as a product of four elements of finite order. In particular, the index map induces a homomorphism from H_1 to K_1 of the groupoid C^*-algebra. Explicit computations of homology groups of AF groupoids and etale groupoids arising from subshifts of finite type are also given.Comment: 34 page