Holomorphic functions on geometrically finite quotients of the ball
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917247231983616 |
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| author | Sarem, William |
| author_facet | Sarem, William |
| contents | Let $Γ$ be a discrete and torsion-free subgroup of $\mathrm{PU}(n,1)$, the group of biholomorphisms of the unit ball in $\mathbb{C}^{n}$, denoted by $\mathbb{H}^{n}_{\mathbb{C}}$. We show that if $Γ$ is Abelian, then $\mathbb{H}^{n}_{\mathbb{C}}/Γ$ is a Stein manifold. If the critical exponent $δ(Γ)$ of $Γ$ is less than 2, a conjecture of Dey and Kapovich predicts that the quotient $\mathbb{H}^{n}_{\mathbb{C}}/Γ$ is Stein. We confirm this conjecture in the case where $Γ$ is parabolic or geometrically finite. We also study the case of quotients with $δ(Γ)=2$ that contain compact complex curves and confirm another conjecture of Dey and Kapovich. We finally show that $\mathbb{H}^{n}_{\mathbb{C}}/Γ$ is Stein when $Γ$ is a parabolic or geometrically finite group preserving a totally real and totally geodesic submanifold of $\mathbb{H}^{n}_{\mathbb{C}}$, without any hypothesis on the critical exponent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16620 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Holomorphic functions on geometrically finite quotients of the ball Sarem, William Complex Variables Differential Geometry Group Theory Geometric Topology Let $Γ$ be a discrete and torsion-free subgroup of $\mathrm{PU}(n,1)$, the group of biholomorphisms of the unit ball in $\mathbb{C}^{n}$, denoted by $\mathbb{H}^{n}_{\mathbb{C}}$. We show that if $Γ$ is Abelian, then $\mathbb{H}^{n}_{\mathbb{C}}/Γ$ is a Stein manifold. If the critical exponent $δ(Γ)$ of $Γ$ is less than 2, a conjecture of Dey and Kapovich predicts that the quotient $\mathbb{H}^{n}_{\mathbb{C}}/Γ$ is Stein. We confirm this conjecture in the case where $Γ$ is parabolic or geometrically finite. We also study the case of quotients with $δ(Γ)=2$ that contain compact complex curves and confirm another conjecture of Dey and Kapovich. We finally show that $\mathbb{H}^{n}_{\mathbb{C}}/Γ$ is Stein when $Γ$ is a parabolic or geometrically finite group preserving a totally real and totally geodesic submanifold of $\mathbb{H}^{n}_{\mathbb{C}}$, without any hypothesis on the critical exponent. |
| title | Holomorphic functions on geometrically finite quotients of the ball |
| topic | Complex Variables Differential Geometry Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2411.16620 |