Timeline for Finitely presented sub-groups of $\operatorname{GL}(n,C)$
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| S May 1, 2023 at 21:09 | history | suggested | The Amplitwist | CC BY-SA 4.0 |
fixed broken link to jlms.oxfordjournals.org; replaced another broken link with WebArchive snapshot; added full citations in tooltips; Drutu -> Druţu
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| May 1, 2023 at 20:36 | review | Suggested edits | |||
| S May 1, 2023 at 21:09 | |||||
| Dec 28, 2009 at 12:59 | comment | added | Pasha Zusmanovich | This Malcev's theorem is explained (nicely, in context of subsequent developments) at least in two places: B.A.F. Wehrfritz, Infinite Linear Groups, Springer, 1973 and A.E. Zalesskii, Linear groups, Russ. Math. Surv. 36 (1981), N5, 63-128. | |
| Dec 26, 2009 at 23:57 | comment | added | HJRW | Malcev's theorem - or at least the assertion that linear groups are residually finite, which is what it amounts to - is also often called "Selberg's Lemma". There's a nice account in Roger Alperin's paper "An elementary account of Selberg's lemma". | |
| Dec 24, 2009 at 6:02 | comment | added | Greg Kuperberg | I conjecture that Dmitri knows Russian and can read the book. Whereas I, with my finely honed language skills, can read the title at a rate of one word per minute. | |
| Dec 24, 2009 at 4:43 | comment | added | Igor Belegradek | A proof of Malcev's theorem is in a book by Merzlyakov "Рациональные группы" (in Russian, the title translates as "Rational groups"). I do not think there is English translation, so probably Malcev's original paper is the most accessible reference. | |
| Dec 24, 2009 at 0:05 | vote | accept | Dmitri Panov | ||
| Dec 24, 2009 at 0:00 | comment | added | Greg Kuperberg | The result was explained to me verbally. I Googled around, but I did not find any better reference than Malcev's original paper, "On the faithful representation of infinite groups by matrices", Mat. Sb. 8 (1940), 405-422. (And Amer. Math. Soc. Transl. (2) 45 (1965) 1-18.) | |
| Dec 23, 2009 at 23:00 | comment | added | Dmitri Panov | Greg, may I ask you one more thing? Is there any good exposition, that decribes the result "$G$ is residually linear if and only if it is residually finite". I like a lot the idea of the proof that you have descirbed, but also want to see it written in more detailes. | |
| Dec 23, 2009 at 20:19 | comment | added | Greg Kuperberg | Provided you are given the faithful linear representations, not just a promise that they exist, the question could be okay. I.e., "Given two subgroups of GL(n,Q) generated by explicit lists of matrices, together with lists of relations and the promise that they are sufficient, is there an algorithm to determine if they are isomorphic as groups?" Note that for closed hyperbolic 3-manifolds, Mostow rigidity is available, which gives much more information than just linearity. | |
| Dec 23, 2009 at 20:11 | comment | added | Dmitri Panov | Greg, thank you for the answer and for the comment to the second part of my question! I guess, I should modify it. I was having in mind the following: It is said that 3-dimensional manifolds are algorithmically recognisible, while 4-manfiolds not, beacause for FPFG groups there is no algorithm to recognise them. So maybe I should reformulate the question like this: Is there an algorithm that recognise linear FPRG groups (i.e. decides for any two such groups if they are isomorphic)? Will this question make more sense? If yes, I will reformulated it like this. | |
| Dec 23, 2009 at 19:28 | history | answered | Greg Kuperberg | CC BY-SA 2.5 |