How to decide whether extensions are quasi isometric

Question

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.

Answer 1

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.