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In group theory, I have seen some results which involve "virtual properties". E. g. virtually abelian, virtually solvable etc. The definition is, according to Wikipedia (https://en.wikipedia.org/wiki/Virtually): A group is virtually $P$ if it contains a subgroup of finite index which has property $P$. Apparently, for example the Tits alternative is an important result which involves "virtually solvable". I have seen some examples of similar theorems which makes be believe that this is a somehow "natural" notion.

Now, what is so important about these "virtual" properties? Why is it more useful then, say, having an infinite (or finite, non trivial) subgroup with the respective property? There is a question about the Tits alternative (The context & motivation for the Tits alternative in combinatorial group theory) where the answerer writes,

"In geometric and combinatorial group theory, "being virtually-$P$" for some property $P$ is basically the same as "being $P$" [...]".

What would be a more precise meaning of this? In what sense are these two properties "basically the same"?

Thank you for your answers:) I could not find an answer on the Internet.

  • Non-abelian free groups are as far from being abelian as you can get, and yet they all contain infinite abelian groups (every element generates an infinite cyclic subgroup). Similarly, $F_2\times \mathbb{Z}_{n}$ contains finite and infinite abelian groups. Having a subgroup of finite index having a property means that "most" of your group has this property. The key phrase is "large-scale geometry", and the key word is "quasi-isometry", which are ideas Gromov really ran with in the 70s-80s and has been important ever since. – user1729 Jan 28 '19 at 09:50
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    HJRW wrote an interesting post on why we care about quasi isometry on MathOverflow. I cannot find it at the moment, but perhaps this different post of his https://mathoverflow.net/a/36406/35478 will help emphasise/explain why "virtual" properties are important. The setting is some old questions in the topology of 3-manifolds (including one of Thurston), and the link is that, roughly, for $M$ a 3-manifold, if $\pi_1(M)$ is virtually-$\mathcal{P}$ then $M$ has a finite sheeted cover whose fundamental group has $\mathcal{P}$. – user1729 Jan 28 '19 at 09:57

2 Answers2

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This isn't a complete answer but it gives one direction to understand why it may be interesting :

In geometric group theory, we look at the Cayley graph of a finitely generated group $G$ on a set of generators $S$. This graph $\Gamma(G,S)$ depends on $S$, but not up to quasi-isometry (it obviously does even up to isometry). In a sense, the quasi-isometry class of all the $\Gamma(G,S)$'s is a good invariant we can associate to $G$ : we are interested in what algebraic properties of $G$ we can deduce from geometric properties of $\Gamma(G,S)$, but these geometric properties "have to be" quasi-isometry invariants.

In particular, if $H$ is a finite index subgroup of $G$, then first of all, since $G$ is finitely generated, so is $H$, but most importantly, "the" Cayley graph of $H$ is quasi-isometric to that of $G$. Hence, any algebraic property we can find on $G$ (resp. $H$) from inspecting only the Cayley graph up to quasi-isometry, we can also find on $H$ (resp. $G$).

In particular, such algebraic properties must be such that $P$ = virtually $P$. Properties with this property are precisely those of the form "virtually $Q$" for some $Q$ (note that "virtually virtually $Q$" is the same as "virtually $Q$").

Thus if we are interested in detecting algebraic properties geometrically, it only makes sense that we have to be interested in virtual properties.

An example of a property that we can detect geometrically is virtual nilpotence, by a celebrated theorem of Gromov.

Maxime Ramzi
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  • Do I understand this correctly: (1) We are interested in properties we can deduce from the Cayley graph. (2) We would be interested in property $P$, but we have no chance to see it from the Cayley graph. (3) So the best approximation is to consider "virtually $P$" instead? –  Jan 23 '19 at 16:35
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    That can be a way to see it, yes. "we have no chance to see it from the Cayley graph" unless it's a virtual property :if a finite index subgroup has it, then the group has it – Maxime Ramzi Jan 23 '19 at 17:59
  • I like your answer but I still don't understand it quite well. I hope you don't mind asking me a follow-up question: Why do we want to see a property from the Cayley graph? I mean, if (1) and (2) from my previous comment would hold, my conclusion was "the Cayley graph is not good in this regard, let's do something else" and not "find something that the Cayley graph can tell us" -- sorry if I am being to naive, but I don't quite understand. –  Jan 24 '19 at 08:38
  • I do like your example with Gromov's theorem. This looks like an explanation to be why virtual nilpotence could be important. –  Jan 24 '19 at 08:51
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    Well that's because it turns out the Cayley graph does tell us some important things; simply it will tell them virtually. Moreover the Cayley graph is a natural invariant and the growth of groups is an interesting topic, in combinatorial and geometric group theory, and this can be directly read off the Cayley graph; so if we're interested in the growth of a group, the Cayley graph pops up and after that we're "stuck" with the virtual properties – Maxime Ramzi Jan 24 '19 at 09:18
  • Hmm, okay. Thank you very much for your effort! I must say I am not 100% satisfied. I will wait with accepting your answer and hope for another answer which may be more complete (you write your answer isn't complete). –  Jan 24 '19 at 14:13
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    Yes of course, don't accept an answer unless you're 100% satisfied – Maxime Ramzi Jan 24 '19 at 14:47
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Without the finite index condition, there's always the trivial subgroup consisting of the identity. So if the definition of "virtual" didn't have the finite index condition, any property P held by the trivial group would also be virtually held by all groups. In particular, the trivial group is solvable, so all groups would be virtually solvable.

Thus, this concept would not be interesting because whether a property is virtually held would not be a property of the group under consideration, but of the trivial group. For each P, either all groups would be virtually P, or none would be.

Acccumulation
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  • Okay, so my example was useless. But why not take "any nontrivial subgroup" or "any infinite or finite subgroup"? I'm sorry if my question was written in a confusing way: I do believe the concept of "virtually" is important (because so many theorems refer to it) but i do not understand why. Sorry, I have right now a hard time writing down what I mean.. do you understand my question? –  Jan 23 '19 at 16:32
  • So I think your answer is more a comment than an answer. Could you thus upgrade it to a comment? –  Jan 25 '19 at 12:23