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

TL;DR

This paper explores how homology groups of almost finite etale groupoids relate to the dynamical properties of their topological full groups, providing explicit computations and structural insights.

Contribution

It establishes a correspondence between homology classes and elements of the topological full group, and analyzes the index map and its kernel, with explicit homology computations.

Findings

Clopen subsets with the same H_0 class are related by the topological full group.

The index map from the topological full group to H_1 is surjective.

Explicit homology calculations for AF groupoids and subshift groupoids.

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 and etale groupoids arising from subshifts of finite type are also given.