Scarparo has constructed counterexamples to Matui's HK-conjecture. These counterexamples are essentially principal but not principal. For group actions the difference between essentially principal and principal is related to the difference between the action being topologically free and free. I will discuss a counterexample to the HK-conjecture that is principal. Like Scarparo's original counterexample, this counterexample is the transformation groupoid associated to a particular odometer. However, the relevant group is the fundamental group of a flat manifold (and hence is torsion-free) and the associated odometer action is free.
A counterexample to the HK-conjecture that is principal