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Suppose we have two extensions of finitely generated groups: $$0 \rightarrow A \rightarrow Q_i \rightarrow W \rightarrow 0 ; i=1,2$$ Is there any way to distinguish not quasi-isometric extensions? At least for simpler cases like $W = Z^{n}$ and $A$ is abelian.

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    For a definition of "quasi-isometry" and some observations that may help you, see [this previous Question](http://math.stackexchange.com/questions/147432/quasi-isometry-geometric-group-theory). – 2017-01-22
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    Maybe my question was not specific enough. What I want to ask is wheter there is some way to classify extensions up to quasi-isometric equivalence – 2017-01-22
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    I imagine you want an algorithmic approach to distinguishing these cases, mainly because of the word "decide" in the title. But it seems necessary to specify what data such algorithms would work on. The short exact(?) sequences shown for $Q_1,Q_2$ give no information that would allow us to distinguish them, much less classify them as not quasi-isometric extensions. – 2017-01-22
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    Well, that was unfortunate word choice, I'd like to know if there is some simpler invariant allowing one to distinguish between non quasi-isometric extensions - something in spirit of classyfing central group extensions with cohomology – 2017-01-22

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This is a very interesting question, but even in the "simpler cases" that you ask about, namely (abelian)-by-(f.g. free abelian) groups, there are no completely general results. And while various special cases are known, those special cases are subtle and the proofs can be hard. While there are some patterns which have led to some generalizations, there's not yet a completely general pattern to be discerned.

Farb and I worked out the case where $A$ is non-finitely-generated abelian, $W$ has rank 1, and $Q$ is finitely presented. See our papers

  • Quasi-isometric rigidity for the solvable Baumslag-Solitar groups. II.
  • On the asymptotic geometry of abelian-by-cyclic groups.

For quasi-isometric rigidity of 3-dimensional solv geometry (i.e. the case $A=\mathbb{Z}^2$, $W=\mathbb{Z}$), plus some other interesting extension groups, look up the paper of Eskin, Fisher, and Whyte:

  • Coarse differentiation of quasi-isometries II: Rigidity for Sol and lamplighter groups

For various cases that allow $W$ to be of higher rank and/or $A$ to be nilpotent, look up papers by Dumarz, Peng, and Xie.

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    Dymarz, not Dumarz – 2017-02-08