Proper actions of Grigorchuk groups on a CAT(0) cube complex
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866912087214653440 |
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| author | Schneeberger, Grégoire |
| author_facet | Schneeberger, Grégoire |
| contents | On this paper we will present a construction of a CAT(0) cube complex (an infinite cube), on which the uncountable family of Grigorchuk groups $G_ω$ act without bounded orbit. Moreover, if the sequence $ω$ does not contain repetition, we prove that the action is proper and faithful. As a consequence of this result, this cube complex is a model for the classifying space of proper actions for all the groups $G_ω$ with $ω$ without repetition. This construction works in a general way for any group acting on a set and which admits a commensurated subset.These examples of non-elliptic actions of infinite finitely generated torsion groups on a non-positively curved cube complex contrast to several established fixed-point theorems concerning actions of torsion groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_04980 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Proper actions of Grigorchuk groups on a CAT(0) cube complex Schneeberger, Grégoire Group Theory 20F65 On this paper we will present a construction of a CAT(0) cube complex (an infinite cube), on which the uncountable family of Grigorchuk groups $G_ω$ act without bounded orbit. Moreover, if the sequence $ω$ does not contain repetition, we prove that the action is proper and faithful. As a consequence of this result, this cube complex is a model for the classifying space of proper actions for all the groups $G_ω$ with $ω$ without repetition. This construction works in a general way for any group acting on a set and which admits a commensurated subset.These examples of non-elliptic actions of infinite finitely generated torsion groups on a non-positively curved cube complex contrast to several established fixed-point theorems concerning actions of torsion groups. |
| title | Proper actions of Grigorchuk groups on a CAT(0) cube complex |
| topic | Group Theory 20F65 |
| url | https://arxiv.org/abs/2207.04980 |