Almost finiteness and homology of certain non-free actions

TL;DR

This paper demonstrates that certain Cantor minimal systems are almost finite, computes their homology groups, and explores the conditions under which their transformation groupoids satisfy the HK conjecture, highlighting the role of freeness.

Contribution

It establishes almost finiteness for specific non-free actions and characterizes when the HK conjecture holds for their groupoids.

Findings

Cantor minimal $bZ timesbZ_2$-systems are almost finite

Homology groups of these systems are computed

HK conjecture holds if and only if the action is free

Abstract

We show that Cantor minimal $\mathbb{Z}\rtimes\mathbb{Z}_2$-systems and essentially free amenable odometers are almost finite. We also compute the homology groups of Cantor minimal $\mathbb{Z}\rtimes\mathbb{Z}_2$-systems and show that the associated transformation groupoids satisfy the HK conjecture if and only if the action is free.