A counterexample to the HK-conjecture that is principal

TL;DR

This paper presents a new principal counterexample to the HK-conjecture using a torsion-free odometer transformation groupoid, expanding understanding of the conjecture's limitations.

Contribution

It provides the first principal counterexample to the HK-conjecture, constructed via a torsion-free odometer groupoid with a free action.

Findings

Counterexample is principal and torsion-free

The example satisfies the rational HK-conjecture

Extends the class of known counterexamples

Abstract

Scarparo has constructed counterexamples to Matui's HK-conjecture. These counterexample and other known counterexamples are essentially principal but not principal. In the present paper, a counterexample to the HK-conjecture that is principal is given. Like Scarparo's original counterexample, our 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. The examples discussed here do satisfy the rational version of the HK-conjecture.