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

Abstract

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 andétale groupoids arising from subshifts of finite type are also given.