Sharpness of proper and cocompact actions on reductive homogeneous spaces
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arXiv
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| Format: | Preprint |
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2024
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| author | Kassel, Fanny Tholozan, Nicolas |
| author_facet | Kassel, Fanny Tholozan, Nicolas |
| contents | We prove that if $G$ is a noncompact connected real reductive linear Lie group, then any discrete subgroup of $G$ acting properly discontinuously and cocompactly on some homogeneous space $G/H$ of $G$ is quasi-isometrically embedded and sharp for $G/H$, i.e. satisfies a strong, quantitative form of proper discontinuity. For noncompact reductive $H$, this was known as the Sharpness Conjecture, with applications to spectral analysis on pseudo-Riemannian locally symmetric spaces developed in arXiv:1209.4075. For $G/H$ rational of real corank one, we use sharpness to fully characterize properly discontinuous and cocompact actions on $G/H$ in terms of Anosov representations. This enables us to show that in real corank one, acting properly discontinuously and cocompactly on $G/H$ is an open property, and also to prove that a number of homogeneous spaces do not admit compact quotients, such as $\mathrm{SL}(n+1,\mathbb{K})/\mathrm{SL}(n,\mathbb{K})$ for $n>1$ and $\mathbb{K}=\mathbb{R}$, $\mathbb{C}$, or the quaternions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_08179 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharpness of proper and cocompact actions on reductive homogeneous spaces Kassel, Fanny Tholozan, Nicolas Group Theory Differential Geometry Geometric Topology We prove that if $G$ is a noncompact connected real reductive linear Lie group, then any discrete subgroup of $G$ acting properly discontinuously and cocompactly on some homogeneous space $G/H$ of $G$ is quasi-isometrically embedded and sharp for $G/H$, i.e. satisfies a strong, quantitative form of proper discontinuity. For noncompact reductive $H$, this was known as the Sharpness Conjecture, with applications to spectral analysis on pseudo-Riemannian locally symmetric spaces developed in arXiv:1209.4075. For $G/H$ rational of real corank one, we use sharpness to fully characterize properly discontinuous and cocompact actions on $G/H$ in terms of Anosov representations. This enables us to show that in real corank one, acting properly discontinuously and cocompactly on $G/H$ is an open property, and also to prove that a number of homogeneous spaces do not admit compact quotients, such as $\mathrm{SL}(n+1,\mathbb{K})/\mathrm{SL}(n,\mathbb{K})$ for $n>1$ and $\mathbb{K}=\mathbb{R}$, $\mathbb{C}$, or the quaternions. |
| title | Sharpness of proper and cocompact actions on reductive homogeneous spaces |
| topic | Group Theory Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2410.08179 |