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We compute the homology groups of transformation groupoids associated with odometers and show that certain
-odometers give rise to counterexamples to the HK conjecture, which relates the homology of an essentially principal, minimal, ample groupoid
with the K-theory of
. We also show that transformation groupoids of odometers satisfy the AH conjecture.
We show that an action of a group on a set X is locally finite if and only if X is not equidecomposable with a proper subset of itself. As a consequence, a group is locally finite if and only if its uniform Roe algebra is finite.
We characterize supramenable groups in terms of the existence of invariant probability measures for partial actions on compact Hausdorff spaces and the existence of tracial states on partial crossed products. These characterizations show that, in general, one cannot decompose a partial crossed product of a
-algebra by a semidirect product of groups into two iterated partial crossed products. However, we give conditions which ensure that such decomposition is possible.
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