Timeline for Is the group of integer points of a simple real linear algebraic group a maximal closed subgroup?
Current License: CC BY-SA 4.0
13 events
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| Oct 15, 2021 at 2:00 | history | edited | Venkataramana | CC BY-SA 4.0 |
last sentence removed
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| Oct 14, 2021 at 18:21 | vote | accept | Ian Gershon Teixeira | ||
| Oct 14, 2021 at 14:47 | comment | added | YCor | @IanGershonTeixeira In the non-simple semisimple case you could have a Zariski-dense maximal discrete group that is not maximal closed (take $SL_n(Z)^2$ in $SL_n(R)^2$), although this can't happen for irreducible lattices (= those with dense projection modulo every simple factor). But in the simple case this equivalence is correct. | |
| Oct 14, 2021 at 13:46 | comment | added | Ian Gershon Teixeira | It seems like this argument proves that for discrete subgroups we have "maximal closed = maximal discrete + zariski dense." Is that correct? | |
| Oct 14, 2021 at 13:16 | comment | added | Ian Gershon Teixeira | Omg I'm so sorry I keep doing that I'll edit it. Oh wait its been too long since I posted I can't edit :( but yes same mental slip really sorry you guys ya I mean maximal closed or equivalently "almost dense" | |
| Oct 14, 2021 at 13:14 | comment | added | LSpice | @IanGershonTeixeira, are you saying dense when you mean maximal closed? | |
| Oct 14, 2021 at 13:11 | history | edited | LSpice | CC BY-SA 4.0 |
`\operatorname` and two spurious spaces
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| Oct 14, 2021 at 13:07 | comment | added | Ian Gershon Teixeira | Ok so there are many options for integral forms. Can you pick one and explain what the integer points look like and show they are not dense in the real points? I love specific examples of groups! | |
| Oct 14, 2021 at 12:54 | comment | added | YCor | @IanGershonTeixeira Note that $G_\mathbf{Z}$ is well-defined only if you fix $G\subset\mathrm{GL}_n$ (or, alternatively, if you define it as scheme over $\mathbf{Z}$), but not starting from $G$ an abstract $\mathbf{R}$-group or even $\mathbf{Q}$-group. In particular, saying $G=\mathrm{SO}_{4,1}$ is highly ambiguous, since you have a lot of choice about the integral form. | |
| Oct 14, 2021 at 12:48 | comment | added | Ian Gershon Teixeira | Wow this helps alot! For the second part of your answer could you give an example of a simple $ G$ with $ G_\mathbb{Z} $ not a maximal closed subgroup in $G_\mathbb{R} $? Can we take $ G$ to be an indefinite signature orthogonal group like $ SO_{4,1} $ perhaps? Can you also give a sense of what the group of integer points look like (is it a free product like modular group etc...) . I think one of the best ways to answer a question is with a specific example of groups! | |
| Oct 14, 2021 at 5:21 | history | edited | Venkataramana | CC BY-SA 4.0 |
typos rectified
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| Oct 14, 2021 at 5:14 | history | edited | Venkataramana | CC BY-SA 4.0 |
typos rectified
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| Oct 14, 2021 at 3:24 | history | answered | Venkataramana | CC BY-SA 4.0 |