Relatively geometric actions of Kähler groups on CAT(0) cube complexes
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917870524432384 |
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| author | Bregman, Corey Groves, Daniel Zhu, Kejia |
| author_facet | Bregman, Corey Groves, Daniel Zhu, Kejia |
| contents | We prove that for $n\geq 2$, a non-uniform lattice in $\text{PU}(n,1)$ does not admit a relatively geometric action on a $\mathrm{CAT}(0)$ cube complex, in the sense of Einstein and Groves. As a consequence, if $Γ$ is a non-uniform lattice in a non-compact semisimple Lie group $G$ without compact factors that admits a relatively geometric action on a $\mathrm{CAT}(0)$ cube complex, then $G$ is commensurable with $\text{SO}(n,1)$. We also prove that if a Kähler group is hyperbolic relative to residually finite parabolic subgroups, and acts relatively geometrically on a $\mathrm{CAT}(0)$ cube complex, then it is virtually a surface group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_12850 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Relatively geometric actions of Kähler groups on CAT(0) cube complexes Bregman, Corey Groves, Daniel Zhu, Kejia Group Theory Complex Variables Geometric Topology 20F65, 22E40 (primary), 32J27, 32J05, 57N65 (secondary) We prove that for $n\geq 2$, a non-uniform lattice in $\text{PU}(n,1)$ does not admit a relatively geometric action on a $\mathrm{CAT}(0)$ cube complex, in the sense of Einstein and Groves. As a consequence, if $Γ$ is a non-uniform lattice in a non-compact semisimple Lie group $G$ without compact factors that admits a relatively geometric action on a $\mathrm{CAT}(0)$ cube complex, then $G$ is commensurable with $\text{SO}(n,1)$. We also prove that if a Kähler group is hyperbolic relative to residually finite parabolic subgroups, and acts relatively geometrically on a $\mathrm{CAT}(0)$ cube complex, then it is virtually a surface group. |
| title | Relatively geometric actions of Kähler groups on CAT(0) cube complexes |
| topic | Group Theory Complex Variables Geometric Topology 20F65, 22E40 (primary), 32J27, 32J05, 57N65 (secondary) |
| url | https://arxiv.org/abs/2210.12850 |