Homology of odometers

We compute the homology groups of transformation groupoids associated with odometers and show that certain infinite dihedral group odometers give rise to counterexamples to the HK conjecture, which relates the homology of an essentially principal, minimal, ample groupoid $G$ with the K-theory of $C_r^{\ast }(G)$. We also show that transformation grupoids of odometers satisfy the AH conjecture.