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PARTIAL ACTIONS OF FREE GROUPS AND GROUPOID HOMOLOGY
- Part of
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- Journal:
- Journal of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 04 September 2026, pp. 1-16
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We give a length-one projective resolution of the trivial module for the groupoid of a semi-saturated partial action (in the sense of Exel) of a free group on a compact Hausdorff and totally disconnected space. As a consequence, we obtain an elementary computation of the homology of these groupoids, which include transformation groupoids of free group actions and Deaconu–Renault groupoids of systems $(X,T)$
, where $X$
is compact Hausdorff and totally disconnected, and $T$
is a local homeomorphism with domain a clopen subset of $X$
. We also show that algebra of such a partial action groupoid over a field has global dimension at most $2$
when the space is second countable.
