Deforming cubulations of hyperbolic groups
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2019
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| _version_ | 1866910066686296064 |
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| author | Fioravanti, Elia Hagen, Mark |
| author_facet | Fioravanti, Elia Hagen, Mark |
| contents | We describe a procedure to deform cubulations of hyperbolic groups by "bending hyperplanes". Our construction is inspired by related constructions like Thurston's Mickey Mouse example, walls in fibred hyperbolic $3$-manifolds and free-by-$\mathbb Z$ groups, and Hsu-Wise turns.
As an application, we show that every cocompactly cubulated Gromov-hyperbolic group admits a proper, cocompact, essential action on a ${\rm CAT}(0)$ cube complex with a single orbit of hyperplanes. This answers (in the negative) a question of Wise, who proved the result in the case of free groups.
We also study those cubulations of a general group $G$ that are not susceptible to trivial deformations. We name these "bald cubulations" and observe that every cocompactly cubulated group admits at least one bald cubulation. We then apply the hyperplane-bending construction to prove that every cocompactly cubulated hyperbolic group $G$ admits infinitely many bald cubulations, provided $G$ is not a virtually free group with ${\rm Out}(G)$ finite. By contrast, we show that the Burger-Mozes examples each admit a unique bald cubulation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_10999 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Deforming cubulations of hyperbolic groups Fioravanti, Elia Hagen, Mark Geometric Topology Group Theory We describe a procedure to deform cubulations of hyperbolic groups by "bending hyperplanes". Our construction is inspired by related constructions like Thurston's Mickey Mouse example, walls in fibred hyperbolic $3$-manifolds and free-by-$\mathbb Z$ groups, and Hsu-Wise turns. As an application, we show that every cocompactly cubulated Gromov-hyperbolic group admits a proper, cocompact, essential action on a ${\rm CAT}(0)$ cube complex with a single orbit of hyperplanes. This answers (in the negative) a question of Wise, who proved the result in the case of free groups. We also study those cubulations of a general group $G$ that are not susceptible to trivial deformations. We name these "bald cubulations" and observe that every cocompactly cubulated group admits at least one bald cubulation. We then apply the hyperplane-bending construction to prove that every cocompactly cubulated hyperbolic group $G$ admits infinitely many bald cubulations, provided $G$ is not a virtually free group with ${\rm Out}(G)$ finite. By contrast, we show that the Burger-Mozes examples each admit a unique bald cubulation. |
| title | Deforming cubulations of hyperbolic groups |
| topic | Geometric Topology Group Theory |
| url | https://arxiv.org/abs/1912.10999 |