Partial actions of free groups and groupoid homology
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914334064508928 |
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| author | Steinberg, Benjamin |
| author_facet | Steinberg, Benjamin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15170 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Partial actions of free groups and groupoid homology Steinberg, Benjamin Operator Algebras Group Theory Rings and Algebras 20J05, 22A22, 16E10 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. |
| title | Partial actions of free groups and groupoid homology |
| topic | Operator Algebras Group Theory Rings and Algebras 20J05, 22A22, 16E10 |
| url | https://arxiv.org/abs/2602.15170 |