Partial actions of free groups and groupoid homology

TL;DR

This paper constructs a projective resolution for the trivial module of groupoids arising from semi-saturated partial actions of free groups, enabling explicit homology computations and revealing low global dimension of associated algebras.

Contribution

It provides a length one projective resolution for the trivial module of these groupoids, facilitating homology calculations and revealing algebraic properties.

Findings

Explicit homology computation for partial action groupoids.

Global dimension of the algebra is at most 2 for second countable spaces.

Includes applications to transformation and Deaconu-Renault groupoids.

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.