Timeline for What finitely presented groups embed into $\operatorname{GL}_2$?
Current License: CC BY-SA 4.0
15 events
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| S Jul 13, 2022 at 22:46 | history | suggested | Samuel Adrian Antz | CC BY-SA 4.0 |
Added \operatorname to GL and SL.
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| Jul 13, 2022 at 20:51 | review | Suggested edits | |||
| S Jul 13, 2022 at 22:46 | |||||
| Jan 2, 2014 at 20:06 | vote | accept | Qiaochu Yuan | ||
| Jul 24, 2012 at 4:42 | comment | added | Ian Agol | @Yves: I forgot about accessibility issues, which can be dealt with in the 3-manifold case familiar to me. I've changed the statement to hold for finitely generated subgroups of $GL_2(K)$, in which case one need worry about only finitely many primes inverted. I think in the general case, one might have to allow entries in the ring of regular functions on the character variety, but I'll have to think about this. The issue is that one can have many more non-archimedean valuations on a function field. | |
| Jul 24, 2012 at 4:37 | history | edited | Ian Agol | CC BY-SA 3.0 |
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| Jul 24, 2012 at 3:34 | comment | added | YCor | @Agol: about your statement about Bass-Serre: how do you check that the process eventually stops? | |
| Jul 23, 2012 at 20:36 | comment | added | Igor Belegradek | Oh, I see that Henry's answer above sheds light on my question. Maybe this is what you meant. | |
| Jul 23, 2012 at 20:33 | comment | added | Ian Agol | I mean one can check that there is not a faithful representation. You enumerate non-trivial elements in the group, and compute whether the representation is non-trivial on each element. The difficulty is to show that something has a faithful rep. | |
| Jul 23, 2012 at 20:25 | comment | added | Igor Belegradek | I am puzzled how given a group with solvable word problem one could check whether there is a faithful representation into $GL_2(\mathbb C)$? Are there specific groups for which the method works other than discrete subgroups of $PSL_2(\mathbb C)$? Say how would it work for the mapping class group? | |
| Jul 23, 2012 at 17:48 | comment | added | Ian Agol | I've deleted the part about unsolvable word problem. | |
| Jul 23, 2012 at 17:45 | history | edited | Ian Agol | CC BY-SA 3.0 |
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| Jul 23, 2012 at 17:44 | comment | added | Ian Agol | @ Igor: good point! I guess what I mean is whether given the presentation, there is a procedure which finds the solution to the word problem. | |
| Jul 23, 2012 at 17:33 | comment | added | Igor Belegradek | @Ian: word problem is solvable for any finitely generated group of matrices with entries in a commutative ring. See e.g. Miller's survey, after theorem 5.1: ms.unimelb.edu.au/~cfm/papers/paperpdfs/msri_survey.all.pdf | |
| Jul 23, 2012 at 16:40 | history | edited | Ian Agol | CC BY-SA 3.0 |
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| Jul 23, 2012 at 16:30 | history | answered | Ian Agol | CC BY-SA 3.0 |