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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)?

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    $\begingroup$ I think this concept is entirely due to Malcev. It has then been considered by Stëpin in the 80's, much before soficity of groups was introduced. It can be useful to some people that a group is approximable by $C$ iff it embeds as a subgroup of some ultraproduct of groups in $C$. $\endgroup$
    – YCor
    Commented Jul 5, 2019 at 16:59
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    $\begingroup$ There are also applications that go in the opposite direction. In other words, one can prove properties about the groups in $C$ by studying $C$-approximable groups (which themselves need not be in $C$). A common example is using pseudofinite groups (which are $C$-approximate if $C$ is the class of finite groups) to prove results about finite groups. $\endgroup$
    – Gabe Conant
    Commented Jul 5, 2019 at 19:04
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    $\begingroup$ Just to be more precise, the notion of soficity invented by Gromov, and also the notion of hyperlinearity essentially due to Connes, are more refined, weaker approximation notions. $\endgroup$
    – YCor
    Commented Jul 9, 2019 at 9:19

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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.

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In general approximation or continuity results for various invariants and functions might be used to extend results from the class $C$ to the class of $C$-approximable groups.

A concrete example is the Determinant Conjecture of W. Lück : it states that for a group $G$ and a matrix $A$ over the group-ring $\mathbb ZG$ its Fuglede--Kadison determinant $\det_{\mathcal NG}(A)$ is $\ge 1$. This has applications to algebraic topology. (For details about this see Lück's book $L^2$-invariants : theory and applications to geometry and $K$-theory, especially Chapter 13.) Note that in general it is not known whether $\det_{\mathcal NG}(A) > 0$.

This conjecture holds for all residually finite groups, and this is established as follows : it is trivial for $G$ finite where the Fuglede--Kadison determinant is essentially the product of singular values (eigenvalues of the symmetrised matrix). Then one proves that in a sequence of finite quotients $G_n$ approximating (sofically) $G$, if the matrices $A_n$ are the images of $A$ we have $$ \limsup_{n\to +\infty} \det{}_{\mathcal NG_n} \le \det{}_{\mathcal NG}(A) $$ and the result follows immediately (this is written up in detail in loc. cit.). I think the argument immediately extends to any sofic approximation though I'm not sure if the details are written up somewhere.

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