Why study "virtual properties"?

Question

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.

Answer 1

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.

Answer 2

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.