Recall the following concept due to Malcev and Gromov. Let $C$ be some class of groups. A group $G$ is said to be approximable by the class $C$ if for every finite symmetric subset $F\subset G$ containing 1, there is a group $H$ in $C$ and an injection $f:F\to H$ such that $f(1)=1$ and $f(x)f(y)=f(z)$ for every $x,y,z\in F$ with $xy=z$.


My question is: why is this concept useful? I.e. what are real application of it in group theory (say, for establishing some properties of $G$ via its approximation by some class)?



What can the approximation of a group by some class be used for?
Let $\mathcal{F}\_k$ be the class of free groups that can be generated by at most $k$ elements. In your terminology, *limit groups* are the groups that can be approximated by the groups in $\mathcal{F}\_k$. [This](https://arxiv.org/abs/math/0401042) article by Champetier--Guirardel develops the theory of limit groups from this point of view. It contains many examples of the kinds of applications that you're looking for.


