Timeline for Hyperbolic 3-manifolds inside algebraic varieties
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Mar 16, 2021 at 17:49 | answer | added | Moishe Kohan | timeline score: 1 | |
| Mar 16, 2021 at 16:08 | answer | added | Toffee | timeline score: 2 | |
| Mar 16, 2021 at 12:36 | comment | added | Nicolast | Yes, I was just suggesting how to construct some examples. Your very detailed comments go far beyond that. You should probably make them an answer? | |
| Mar 16, 2021 at 1:02 | comment | added | Toffee | Lastly, there are hyperbolic 3-manifolds that cannot be a totally geodesic subspace of any complex hyperbolic n-manifold (for any n). See this post by Ian Agol: Lastly, also see this post by Ian Agol: mathoverflow.net/questions/281650/… | |
| Mar 16, 2021 at 0:59 | comment | added | Toffee | The complex hyperbolic lattices here don't have CSP, but the embedding question is subtle (and very interesting, IMO). Not to say you didn't know this already @NicolasTholozan, just adding this to the discussion. Moreover, it would be interesting to say something stronger: which hyperbolic 3-manifolds are the fixed set of an antiholomorphic involution of a complex hyperbolic 3-fold. | |
| Mar 16, 2021 at 0:56 | comment | added | Toffee | This is the same as asking for the profinite topology on $\Gamma$ to induce the full profinite topology on $\Lambda$. This isn't always the case. For example, for $\Gamma = \mathrm{SL}_3(\mathbb{Z})$, you have $\mathrm{SL}_2(\mathbb{Z})$ inside. However $\mathrm{SL}_3(\mathbb{Z})$ has the congruence subgroup property, so the only subgroups of $\mathrm{SL}_2(\mathbb{Z})$ that embed totally geodesically in a finite volume quotient of $\mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}(3)$ (via this embedding) are the congruence covers of the modular surface. | |
| Mar 16, 2021 at 0:53 | comment | added | Toffee | The problem is that certain elements of the commensurability class might not be a maximal $\mathbb{H}^3$ stabilizer inside a lattice. You start with $\Lambda = \mathrm{SO}(q, \mathcal{O}_K)$ inside $\Gamma = \mathrm{SU}(q, \mathcal{O}_L)$ for $L/K$ a totally imaginary extension of a totally real field. Given an arbitrary $\Lambda^\prime$ in $\Lambda$ of finite index, I need to know there's $\Gamma^\prime < \Gamma$ of finite index so that $\Gamma^\prime \cap \Lambda = \Lambda^\prime$. | |
| Mar 16, 2021 at 0:48 | comment | added | Toffee | @NicolasTholozan It's clear that many arithmetic hyperbolic 3-manifolds are commensurable with a totally geodesic real subspace of an arithmetic quotient of complex hyperbolic 3-space, but realizing everything in the commensurability class is, to my knowledge, an open problem. | |
| Mar 15, 2021 at 22:35 | comment | added | Nicolast | Many arithmetic hyperbolic $3$-manifolds embed as a totally geodesic real subspace of an arithmetic quotient of the complex ball in $\mathbb C^3$ (hence a complex Shimura variety). | |
| Mar 15, 2021 at 21:19 | comment | added | Moishe Kohan | Some (closed or finite volume) hyperbolic 3-manifolds one can embed in complex-projective surfaces. I very much doubt that this is always possible. All hyperbolic 3-manifolds fibered over $S^1$ admit natural (canonical) embeddings in certain noncompact Kahler surfaces. | |
| Mar 15, 2021 at 20:33 | comment | added | Sam Nead | When is a three-manifold the real locus of a three-dimensional complex algebraic variety? | |
| Mar 15, 2021 at 19:57 | history | edited | Afa | CC BY-SA 4.0 |
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| Mar 15, 2021 at 19:56 | comment | added | YCor | In the context embeddings of surfaces into hyperbolic 3-manifolds, there is no higher $\pi_i$, thus $\pi_1$-injectivity would sound more relevant in the absence of higher homotopy in the larger variety. | |
| Mar 15, 2021 at 19:53 | history | edited | Afa | CC BY-SA 4.0 |
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| Mar 15, 2021 at 19:24 | answer | added | riccoli | timeline score: 5 | |
| Mar 15, 2021 at 19:23 | comment | added | Will Sawin | Just a thought on what "beauty" might mean: The condition "injective on $\pi_1$" is a good one for embedding surfaces in hyperbolic. $3$-manifolds, so maybe it is also a good condition for embedding hyperbolic $3$-manifolds in algebraic varieties. | |
| Mar 15, 2021 at 19:05 | history | edited | YCor |
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| Mar 15, 2021 at 18:51 | history | asked | Afa | CC BY-SA 4.0 |