Timeline for Universal group?
Current License: CC BY-SA 4.0
29 events
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| Aug 31, 2022 at 10:04 | comment | added | The Amplitwist |
Another link to springerlink.com in a comment above seems to be broken.
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| Jun 8, 2022 at 7:22 | comment | added | The Amplitwist |
@AntonPetrunin The link to springerlink.com is broken. Perhaps you could take a look, whenever possible...
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| May 3, 2022 at 8:33 | comment | added | YCor | "universal" seems to become a universal terminology... | |
| May 3, 2022 at 8:01 | history | edited | Glorfindel | CC BY-SA 4.0 |
broken link fixed
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| Mar 5, 2014 at 4:41 | comment | added | მამუკა ჯიბლაძე | It is probably self-understood but the words "torsion free" are missing, right? | |
| Mar 5, 2014 at 3:36 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
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| Mar 22, 2012 at 19:09 | comment | added | Anton Petrunin | @YangMills, I think "all-inclusive" describes the property better, although "telescopic" sounds better. | |
| Mar 22, 2012 at 19:03 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
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| Mar 22, 2012 at 16:41 | comment | added | YangMills | How come now "telescopic" has become "all-inclusive"? | |
| Oct 9, 2011 at 18:45 | vote | accept | Anton Petrunin | ||
| Oct 9, 2011 at 2:48 | answer | added | Anton Petrunin | timeline score: 6 | |
| Oct 9, 2011 at 1:09 | comment | added | Greg Kuperberg | @Alan @Mark @Igor @HW @Agol @Anton Could someone write in an answer to move the question to answered status? | |
| Sep 17, 2011 at 3:18 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
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| Mar 29, 2011 at 16:55 | history | edited | Anton Petrunin | CC BY-SA 2.5 |
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| Dec 17, 2010 at 8:48 | comment | added | ADL | Such a group is necessarily Large (Large in the sense of Gromov/Pride) also (this implies SQ-universal, along with a number of other properties). See this paper of Button for more details, springerlink.com/content/u014p6u2w4560786 | |
| Dec 17, 2010 at 1:36 | history | edited | Anton Petrunin | CC BY-SA 2.5 |
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| Oct 5, 2010 at 0:46 | comment | added | user6976 | Anton: Am I right and your group is given by the presentation $\langle x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3, t_1, t_2, t_3| u^2=1 \hbox{(for all generators)}, x_1x_2x_3y_1y_2y_3=z_1z_2z_3, x_1x_2x_3(y_1y_2y_3)^{-1}=t_1t_2t_3\rangle$ ? Such groups have not been constructed before. It is not possible to call them "universal" for the reasons given to you by Igor. The term has been used already for something else. | |
| Mar 31, 2010 at 21:21 | comment | added | Igor Belegradek | Anton, usually for a "universal group" the input is a finite presentation, for example, the above mentioned universal finitely presented group starts with the free product of all finitely presented groups, which is recursively presented, and then embeds it into a finitely presented group. Similarly, in a Rips construction one starts with a finite presentation and adds/modifies relations. I just wish to see where all finite presentations come in? | |
| Mar 31, 2010 at 20:08 | comment | added | Anton Petrunin | @Igor. It is a direct construction, but it takes about 2-3 pages to write... | |
| Mar 31, 2010 at 19:21 | comment | added | Igor Belegradek | @Anton, would you explain what makes the group universal? | |
| Mar 31, 2010 at 18:09 | comment | added | Anton Petrunin | @Igor, Take a figure eight with generators $a$ and $b$. Glue four 2-cells along words $a$, $b$, $ab$ and $ab^{-1}$. (The obtained space is simply connected.) Now, choose 3 points in each 2-cell and make each to be a double orbi-point. The obtained orbi-space has needed fundamental group. | |
| Mar 31, 2010 at 15:22 | comment | added | Igor Belegradek | @Anton, would you tells us how the group $G$ is constructed? | |
| Mar 31, 2010 at 3:07 | comment | added | Igor Belegradek | Anton, by a "universal finitely presented group" one usually means a finitely presented group that contains each finitely presented group as a subgroup. Such groups can be constructed via Higman's embedding theorem. If $Q$ is such a group, it is possible to cook up a hyperbolic group $G$ such that $Q$ is a quotient of $G$, and the kernel is normally generated by elements of finite order. This is of course not the same as what you do. | |
| Mar 31, 2010 at 1:41 | comment | added | HJRW | This property seems far too specific to be called simply "universal". I'd go with something like "TQ-universal". If you want to know whether someone else has done this, I'd try looking at the work of Olshanskii and his students. | |
| Mar 31, 2010 at 1:33 | comment | added | Ian Agol | The closest condition I've heard of is "SQ-universal": en.wikipedia.org/wiki/SQ_universal_group Your group satisfies a very strong form of "SQ-universal in the class of finitely presented groups". | |
| Mar 31, 2010 at 1:26 | comment | added | Theo Johnson-Freyd | I think "universal finitely-presented group" is OK: you claim that any finitely presented group is a subquotient, which although not quite a universal property en.wikipedia.org/wiki/Universal_property#Formal_definition isn't too far removed. But certainly there are non-finitely-presented groups that you're not capturing. | |
| Mar 31, 2010 at 0:34 | comment | added | Anton Petrunin | Well, Swiss-Army Group is a nice name. But why not universal? --- after quick search I did not see that term "universal group" is used... | |
| Mar 30, 2010 at 23:48 | comment | added | Gerhard Paseman | I advise against the word "universal", without more context at least. Call it Anton-universal or the Petrunin-Swiss-Army Group, or some useful modification of some synonym for "universal". Gerhard "Ask Me About System Design" Paseman, 2010.03.30 | |
| Mar 30, 2010 at 23:30 | history | asked | Anton Petrunin | CC BY-SA 2.5 |