Timeline for What finitely presented groups embed into $\operatorname{GL}_2$?
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| S Jul 13, 2022 at 21:32 | history | suggested | Samuel Adrian Antz | CC BY-SA 4.0 |
Added \operatorname to GL.
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| Jul 13, 2022 at 20:48 | review | Suggested edits | |||
| S Jul 13, 2022 at 21:32 | |||||
| Jan 2, 2014 at 20:06 | vote | accept | Qiaochu Yuan | ||
| Aug 1, 2012 at 2:06 | comment | added | Qiaochu Yuan | @Agol: thanks! It seems quite interesting. | |
| Jul 31, 2012 at 17:19 | comment | added | Ian Agol | There's this book (which is essentially about computing $SL_2$ character varieties of groups): books.google.com/… | |
| Jul 31, 2012 at 14:34 | answer | added | Misha | timeline score: 6 | |
| Jul 24, 2012 at 6:28 | answer | added | YCor | timeline score: 14 | |
| Jul 24, 2012 at 0:30 | answer | added | Geoff Robinson | timeline score: 3 | |
| Jul 23, 2012 at 21:42 | comment | added | Igor Belegradek | @Qiaochu: look at question 1.2 and theorem 1.3 of archive.numdam.org/article/ASENS_1998_4_31_3_329_0.pdf. Note that tameness is no longer a conjecture, see en.wikipedia.org/wiki/Tameness_theorem. | |
| Jul 23, 2012 at 21:25 | comment | added | Qiaochu Yuan | (Sorry, that should say "properly contained in $H$.") Anyway, I guess I am wondering how weird a f.p. subgroup of $\text{GL}_2(\mathbb{C})$ can get (that is, to what extent I can construct such a group with various prescribed properties). | |
| Jul 23, 2012 at 21:19 | comment | added | Qiaochu Yuan | @Igor: it seems like a natural question to ask to me. I am pretty sure I understand $1$-dimensional representations of groups; I am trying to figure out to what extent I understand $2$-dimensional representations. A more specific motivation is that in an intro abstract algebra course I am grading, the lecturer asked for examples of a group $G$ with a subgroup $H$ such that some conjugate of $H$ is properly contained in $G$. At first I thought one could not construct an example using methods the students had seen, but I later realized that there was an example of such a $G$ in $\text{GL}_2$... | |
| Jul 23, 2012 at 20:10 | comment | added | Igor Belegradek | The easiest thing is to consider fundamental groups of hyperbolic 3-manifolds, and analyse when their holonomy lifts from $PSL_2(\mathbb C)$ to $GL_2(\mathbb C)$. The latter involves obstruction theory considerations. On the other hand it is clear that many (most?) fp groups do not embed into $GL_2(\mathbb C)$. It would help if you could provide some motivation and explain what groups you care about. | |
| Jul 23, 2012 at 19:55 | answer | added | HJRW | timeline score: 12 | |
| Jul 23, 2012 at 16:30 | answer | added | Ian Agol | timeline score: 17 | |
| Jul 23, 2012 at 15:52 | comment | added | Qiaochu Yuan | @Mrc: yes, I am aware of that. My issue is that I already don't know what finitely presented groups appear as nice subgroups of $\text{GL}_2(\mathcal{O}_K)$. | |
| Jul 23, 2012 at 15:07 | comment | added | Benjamin Steinberg | A minimum necessary condition is residual finiteness. But it should be even stronger. The group must live in GL_2 of a finitely generated ring. Such groups somehow are close to being free. For example GL_2(Z) is virtually free. Experts will say more. | |
| Jul 23, 2012 at 15:04 | comment | added | Marc Palm | You can always embed $O_K$ into the complex numbers, so that will do for $GL(2,O_K)$, or not? | |
| Jul 23, 2012 at 14:58 | history | asked | Qiaochu Yuan | CC BY-SA 3.0 |