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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2410.17107 |
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| _version_ | 1866916448915423232 |
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| author | Schwermer, Joachim |
| author_facet | Schwermer, Joachim |
| contents | Using the quaternionic formalism for the description of the group of isometries of hyperbolic $5$-space we consider arithmetically defined $5$-dimensional hyperbolic manifolds which are non-compact but of finite volume. They arise from maximal orders $Λ$ in the central simple algebra $M_2(D)$ of degree $4$ where $D$ denotes a definite quaternion $\mathbb{Q}$-algebra. The affine $\mathbb{Z}$-group scheme $SL_Λ$ determines an integral structure for the algebraic $\mathbb{Q}$-group $G = SL_Λ \times_{\mathbb{Z}} \mathbb{Q}$ obtained by base change. The group $G$ is an inner form of the special linear $\mathbb{Q}$-group $SL_4$. Each torsion-free subgroup $Γ\subset SL_Λ(\mathbb{Z})$ determines a hyperbolic $5$-manifold, to be denoted $X_G/Γ$. Given a principal congruence subgroup $Γ(\frak{p}^e)$, we determine the number of ends and the dimensions of the cohomology groups at infinity of the manifold $X_G/Γ(\frak{p}^e)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17107 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On arithmetically defined hyperbolic $5$-manifolds arising from maximal orders in definite $\mathbb{Q}$-algebras Schwermer, Joachim Number Theory Algebraic Topology 11F75, 57K50, 11R52 Using the quaternionic formalism for the description of the group of isometries of hyperbolic $5$-space we consider arithmetically defined $5$-dimensional hyperbolic manifolds which are non-compact but of finite volume. They arise from maximal orders $Λ$ in the central simple algebra $M_2(D)$ of degree $4$ where $D$ denotes a definite quaternion $\mathbb{Q}$-algebra. The affine $\mathbb{Z}$-group scheme $SL_Λ$ determines an integral structure for the algebraic $\mathbb{Q}$-group $G = SL_Λ \times_{\mathbb{Z}} \mathbb{Q}$ obtained by base change. The group $G$ is an inner form of the special linear $\mathbb{Q}$-group $SL_4$. Each torsion-free subgroup $Γ\subset SL_Λ(\mathbb{Z})$ determines a hyperbolic $5$-manifold, to be denoted $X_G/Γ$. Given a principal congruence subgroup $Γ(\frak{p}^e)$, we determine the number of ends and the dimensions of the cohomology groups at infinity of the manifold $X_G/Γ(\frak{p}^e)$. |
| title | On arithmetically defined hyperbolic $5$-manifolds arising from maximal orders in definite $\mathbb{Q}$-algebras |
| topic | Number Theory Algebraic Topology 11F75, 57K50, 11R52 |
| url | https://arxiv.org/abs/2410.17107 |