Timeline for Examples of locally compact groups that do not admit enough finite dimensional representations
Current License: CC BY-SA 4.0
16 events
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| Dec 13, 2021 at 6:16 | comment | added | Izaak Meckler | Thanks Matt -- fixed. | |
| Dec 13, 2021 at 6:16 | history | edited | Izaak Meckler | CC BY-SA 4.0 |
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| Dec 12, 2021 at 0:55 | comment | added | Matt Zaremsky | Just pointing out that you mean Thompson's group $T$ or $V$, not $F$ ($F$ is not simple). | |
| Dec 10, 2021 at 13:35 | vote | accept | Rick Sternbach | ||
| Dec 10, 2021 at 13:35 | |||||
| Dec 8, 2021 at 19:49 | history | edited | Izaak Meckler | CC BY-SA 4.0 |
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| Dec 8, 2021 at 19:40 | history | edited | Izaak Meckler | CC BY-SA 4.0 |
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| Dec 8, 2021 at 8:31 | comment | added | YCor | (...) Malcev's result is proved by showing that every f.g. domain is residually a finite field. Once this is known, it follows that every f.g. group having a non-trivial $n$-dimensional representation over some field, also has a non-trivial $n$-dimensional representation over some finite field. | |
| Dec 8, 2021 at 8:28 | comment | added | YCor | Sorry, indeed I didn't see you assumed to be finite and thought you were implicitly using Malcev's theorem that f.g. subgroups of $GL_n$ over any field, are residually finite. So my point is that a finitely generated group with no nontrivial finite quotient has no nontrivial (finite-dim) linear representation over any field. By the way, Malcev's result is classical and finite generation is enough (no finite presentability is needed). | |
| Dec 7, 2021 at 19:06 | comment | added | Izaak Meckler | By the first assertion do you mean “an infinite simple group G has no non-trivial representations in finite fields”? This is true because the image is a finite quotient of G of which there is only the trivial group. I further imposed the group should be finitely presented in the second paragraph | |
| Dec 7, 2021 at 19:00 | history | edited | Izaak Meckler | CC BY-SA 4.0 |
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| Dec 7, 2021 at 8:29 | comment | added | YCor | "infinite simple" should be "finitely generated infinite simple". Otherwise $\mathrm{SL}_3(\mathbf{Q})$ is a counterexample to the first assertion. | |
| Dec 7, 2021 at 5:27 | history | edited | Izaak Meckler | CC BY-SA 4.0 |
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| Dec 7, 2021 at 2:43 | history | edited | Izaak Meckler | CC BY-SA 4.0 |
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| Dec 7, 2021 at 2:13 | history | edited | Izaak Meckler | CC BY-SA 4.0 |
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| Dec 7, 2021 at 2:06 | history | edited | Izaak Meckler | CC BY-SA 4.0 |
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| Dec 7, 2021 at 1:48 | history | answered | Izaak Meckler | CC BY-SA 4.0 |