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 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.

Tags:

Gr.Group Theory

Model Theory

Lo.Logic

Algebraic Groups

Finite Groups

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