Timeline for When is a unitary group over a ring of integers dense?
Current License: CC BY-SA 4.0
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| Jul 26, 2023 at 20:47 | comment | added | Aurel | Well, the integral unitary group is the group of unitary matrices that have integral coefficients... Every arithmetic group is discrete when embedded in the correct product of Lie groups, so if this product is compact you get a finite group. No, arithmetic groups can also be dense, if you omit an embedding into a Lie group that is not compact. | |
| Jul 26, 2023 at 15:15 | comment | added | Ian Gershon Teixeira | @Aurel I see, then I need to learn more about S-arithmetic groups. Can you explain more what the (proper) definition of the integral unitary group is and why it must be finite? Is it generally true that an arithmetic subgroup of a compact group is always finite? And only S-arithmetic subgroups can be dense? | |
| Jul 26, 2023 at 9:01 | comment | added | Aurel | The integral unitary group is always finite. In the dense example you give, the first matrix is not integral. If you extend the question to S-integral conditions, the answer is yes by the strong approximation theorem for S-arithmetic groups. | |
| Jul 25, 2023 at 22:26 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| Jul 25, 2023 at 18:01 | history | edited | Ian Gershon Teixeira | CC BY-SA 4.0 |
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| Jul 25, 2023 at 15:46 | history | asked | Ian Gershon Teixeira | CC BY-SA 4.0 |