Homology and K-theory for self-similar actions of groups and groupoids
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916994048065536 |
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| author | Miller, Alistair Steinberg, Benjamin |
| author_facet | Miller, Alistair Steinberg, Benjamin |
| contents | Nekrashevych associated to each self-similar group action an ample groupoid and a $\mathrm{C}^\ast$-algebra. We perform complete computations of the homology of the groupoid and the K-theory of the $\mathrm{C}^\ast$-algebra for a myriad of examples, including the Grigorchuk group, the Grigorchuk--Erschler group, Gupta--Sidki groups, and self-similar actions of free abelian groups and lamplighter groups. The key development is the construction, for arbitrary self-similar group actions, of long exact sequences which compute the homology and K-theory in terms of the homology of the group and K-theory of the group $\mathrm{C}^\ast$-algebra via the transfer map and the virtual endomorphism. Results are proved more generally for self-similar groupoids. As a consequence of our results and recent results of X.~Li, we are able to show that Röver's simple group containing the Grigorchuk group and Thompson's group $V$ is rationally acyclic but has nontrivial Schur multiplier. We prove many more Röver--Nekrashevych groups of self-similar groups are rationally acyclic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02359 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homology and K-theory for self-similar actions of groups and groupoids Miller, Alistair Steinberg, Benjamin Operator Algebras Dynamical Systems Group Theory K-Theory and Homology 20J05, 22A22, 46L05, 46L80, 20F65 Nekrashevych associated to each self-similar group action an ample groupoid and a $\mathrm{C}^\ast$-algebra. We perform complete computations of the homology of the groupoid and the K-theory of the $\mathrm{C}^\ast$-algebra for a myriad of examples, including the Grigorchuk group, the Grigorchuk--Erschler group, Gupta--Sidki groups, and self-similar actions of free abelian groups and lamplighter groups. The key development is the construction, for arbitrary self-similar group actions, of long exact sequences which compute the homology and K-theory in terms of the homology of the group and K-theory of the group $\mathrm{C}^\ast$-algebra via the transfer map and the virtual endomorphism. Results are proved more generally for self-similar groupoids. As a consequence of our results and recent results of X.~Li, we are able to show that Röver's simple group containing the Grigorchuk group and Thompson's group $V$ is rationally acyclic but has nontrivial Schur multiplier. We prove many more Röver--Nekrashevych groups of self-similar groups are rationally acyclic. |
| title | Homology and K-theory for self-similar actions of groups and groupoids |
| topic | Operator Algebras Dynamical Systems Group Theory K-Theory and Homology 20J05, 22A22, 46L05, 46L80, 20F65 |
| url | https://arxiv.org/abs/2409.02359 |