Partial actions of free groups and groupoid homology
Benjamin Steinberg
TL;DR
The paper addresses computing the homology of groupoids arising from semi-saturated partial actions of free groups on compact totally disconnected spaces by constructing an explicit length-one projective resolution of the trivial module. This enables direct calculations yielding $H_n(F_A\ltimes X)=0$ for $n\ge 2$ and explicit descriptions of $H_0$ and $H_1$, and it establishes a global-dimension bound of $2$ for the associated groupoid algebras over a field when the space is second countable. The results apply to Deaconu–Renault groupoids and provide a streamlined, elementary approach to homology and cohomology in this class, including a spectral-sequence argument linking Ext with groupoid cohomology. The work also conjectures stronger structural properties (e.g., hereditary algebras) and connects to broader themes such as the Baum–Connes and HK conjectures via the described cohomological dimensions.
Abstract
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.