Timeline for Geometric or topological results from group theory
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Sep 26, 2021 at 17:52 | history | edited | Stefan Kohl♦ |
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| Sep 26, 2021 at 17:51 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| May 12, 2016 at 4:37 | comment | added | ThiKu | For the proof of Milnor's theorem, I think one can phrase it without mentioning groups: any two metrics on a compact surface are quasi-isometric, so are their lifts to the universal covering space. So the volume growth of metric balls has to be of the same type (exponential or polynomial) in both cases. So one can not have both, a flat metric and a negatively curved one. The important concept here is that of quasi-isometry, while the fundamental group and the Milnor-Svarc lemma are actually not necessary. | |
| May 11, 2016 at 20:58 | answer | added | HJRW | timeline score: 3 | |
| May 11, 2016 at 16:22 | comment | added | Seirios | As you said, now group theory is intertwined with geometry. I am not looking for proofs which use only group theory, but rather proofs where the introduction of a notion of group theory is fundamental, and a fortiori unexpected. For the example of the Virtual Haken conjecture, the key point is to prove the separability of some subgroups in the fundamental group, so for me this is a good example. | |
| May 10, 2016 at 8:36 | comment | added | HJRW | @Seiros, I'm still puzzled by what you mean by 'group theory'. For instance, Benjamin Steinberg mentioned the Virtual Haken conjecture, and indeed, the theorem 'any cubulable hyperbolic group is a quasiconvex subgroup of a right-angled Artin group' sounds a lot like group theory. On the other hand, the theorem 'any npc cube complex with hyperbolic fundamental group is virtually special' sounds less like group theory, and indeed Agol's proof (and Wise's previous work) is phrased more in the language of cube complexes. So do these qualify or not? | |
| May 10, 2016 at 7:35 | comment | added | Seirios | Let us replace "proved using group theory" with "proved using surprisingly group theory". I mean a statement which seems not to be connected to group theory at first glance, but nevertheless which can be proved by introducing groups in the good place. Of course, we do not allow the arguments which turn out to be completely elementary nowadays: for instance, the fundamental group as a tool to distinguish spaces or knots. | |
| May 9, 2016 at 20:32 | comment | added | HJRW | I'm not convinced that any such results are proved purely using group theory: certainly none of the examples given so far qualify. The fact of the matter is that modern group theory is inextricably intertwined with geometry. If the question is just 'list theorems in geometric and topological group theory', then it's far too broad | |
| May 9, 2016 at 19:57 | comment | added | Benjamin Steinberg | I think the virtual Haken and virtually Fibering conjectures are good examples where group theory plays a key role although there are other techniques as well. | |
| May 9, 2016 at 18:14 | comment | added | eins6180 | @ThiKu: The same is actually true for Bieberbach's theorem. It also doesn't really use group theory. | |
| May 9, 2016 at 17:46 | comment | added | ThiKu | I'm not sure whether I would call Milnor's theorem a "result proved by group theory". I think the proof is pure geometry. | |
| May 9, 2016 at 17:20 | comment | added | Steve Huntsman | Do, e.g., results concerning word-hyperbolic groups count, or are they disqualified on grounds that the spaces involved are "synthesized" by the groups themselves? | |
| May 9, 2016 at 17:13 | answer | added | Steve Huntsman | timeline score: 16 | |
| May 9, 2016 at 17:03 | history | asked | Seirios | CC BY-SA 3.0 |