Timeline for Residual finiteness: why do we care?
Current License: CC BY-SA 3.0
26 events
| when toggle format | what | by | license | comment | |
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| Dec 29, 2017 at 6:39 | answer | added | Ian Agol | timeline score: 5 | |
| Mar 4, 2015 at 2:53 | answer | added | Venkataramana | timeline score: 3 | |
| Mar 4, 2015 at 2:47 | answer | added | Yago Antolin | timeline score: 8 | |
| Mar 2, 2015 at 17:13 | answer | added | Igor Rivin | timeline score: 5 | |
| Feb 28, 2015 at 4:30 | answer | added | HJRW | timeline score: 19 | |
| Feb 27, 2015 at 22:19 | history | edited | Geordie Williamson | CC BY-SA 3.0 |
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| Feb 27, 2015 at 18:12 | comment | added | YCor | I'm wondering why the anonymous user who asked the question claims that the late sixties were the golden age of group theory. It certainly was the golden age of the quest of finite simple groups but this is not the end of the story for finite groups, let alone infinite groups. I voted to close this question as primarily opinion-based (and found it essentially pointless), but it was reopened. It would be nice if people who found this question of mathematical interest could edit it so as to make it less polemical. | |
| Feb 27, 2015 at 17:49 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Feb 27, 2015 at 17:48 | comment | added | Jean Raimbault | About 2. and hyperbolic groups: if an hyperbolic group is residually finite then it has a finite index subgroups whose group algebra (over any field) has no units besides the obvious ones (this is a theorem of T. Delzant). | |
| Feb 27, 2015 at 17:36 | history | reopened |
David E Speyer Joonas Ilmavirta Kevin Walker Paul Taylor Andrey Rekalo |
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| Feb 27, 2015 at 15:20 | review | Reopen votes | |||
| Feb 27, 2015 at 17:36 | |||||
| Feb 27, 2015 at 4:16 | comment | added | Ian Agol | Residual finiteness (or rather a strengthening) was used by Friedl and Vidussi to prove: a closed 3-manifold x S^1 is symplectic iff the 3-manifold fibers over the circle. As for hyperbolic groups, there are various equivalent conditions to residual finiteness, such as every hyperbolic group has a finite-index torsion-free subgroup. This may be of interest, for example, to understand cohomological dimension, or to get rid of finite center. | |
| Feb 27, 2015 at 1:33 | comment | added | HJRW | A proof that all hyperbolic groups are residually finite would be a major step forward in our understanding of the topology of hyperbolic spaces. Though for some reason, the 'big picture' for the OP only seems to involve algebra... | |
| Feb 26, 2015 at 22:50 | history | closed |
Alex Degtyarev YCor Stefan Kohl♦ Peter Crooks Qiaochu Yuan |
Opinion-based | |
| Feb 26, 2015 at 20:12 | answer | added | Francesco Polizzi | timeline score: 44 | |
| Feb 26, 2015 at 20:10 | comment | added | Benjamin Steinberg | Profinite groups are precisely the residually finite groups in the category of compact groups (meaning the finite continuous images separate points) and at least these have interest in number theory. The Grothendieck fundamental group of certain varieties is sometimes a profinite completion of a residually finite group. | |
| Feb 26, 2015 at 20:07 | comment | added | Benjamin Steinberg | For a finitely generated group, it is equivalent to being embeddable in a compact group. In general, it is equivalent to having a spherically transitive action on a rooted tree. | |
| Feb 26, 2015 at 20:07 | comment | added | Benjamin Steinberg | Residual finiteness is a topological property. If roughly means that if $G=\pi_1(X)$ and $C$ is a compact subspace of the universal cover $\tilde X$ of $X$, then there is a finite cover $Y$ of $X$ such that the map $\tilde X\to Y$ is injective on $C$. (Here $X$ should be nice enough.) | |
| Feb 26, 2015 at 19:54 | review | Close votes | |||
| Feb 26, 2015 at 22:54 | |||||
| Feb 26, 2015 at 19:47 | answer | added | Gerhard Paseman | timeline score: 2 | |
| Feb 26, 2015 at 19:43 | history | edited | user68579 | CC BY-SA 3.0 |
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| Feb 26, 2015 at 19:39 | comment | added | Yiftach Barnea | I think it would be better to formulate the question not as why do we care but as what are the implications. | |
| Feb 26, 2015 at 19:37 | comment | added | Yemon Choi | I think this is a fair question. Not all of us are group theorists, and even those of us who are might not spend much time pondering whether things are residually finite. (As a kind of harmonic analyst, I am interested in whether certain groups are residually finite, but that's a side-note) | |
| Feb 26, 2015 at 19:34 | comment | added | Alex Degtyarev | The next natural question would by why that theorem in number theory that holds is so amazing, or why the property of the class of rings is wonderful. | |
| Feb 26, 2015 at 19:31 | review | First posts | |||
| Feb 26, 2015 at 19:56 | |||||
| Feb 26, 2015 at 19:27 | history | asked | user68579 | CC BY-SA 3.0 |