Are extensions quasi-isometric

Question

Let $0 \rightarrow K \rightarrow G_1 \rightarrow L \rightarrow 0 $ $$$$ $0 \rightarrow K \rightarrow G_2 \rightarrow L \rightarrow 0 $ $$$$ be two group extensions with $G_1,G_2$ finitely generated. Are $G_1$ and $G_2$ necessarily quasi-isometric (with respect to their word metric)?Thank you for all your answers.

Answer 1

No, they need not be quasi-isometric.

For example, using $K=\mathbb{Z}^2$ and $L=\mathbb{Z}$, the extension group $G$ is a semidirect product where the action of $L$ on $K$ is defined by some $M \in GL(2,\mathbb{Z})$. In this case there are three quasi-isometry classes (as one knows from studying the 8 geometries of 3-manifolds, due to W. Thurston):

1. If $|\text{trace}(M)| < 2$, i.e. if $M$ has finite order, then $G$ is quasi-isometric to $\mathbb{R}^3$.
2. If $|\text{trace}(M)| = 2$, then $G$ is quasi-isometric to 3 dimensional NIL geometry.
3. If $|\text{trace}(M)| > 2$, then $G$ is quasi-isometric to 3-dimensional SOLV geometry.

Things will get even wilder for more complicated kernels and quotients.

Answer 2

Let's Take $G_{1}$ = $HF_{3}(\mathbb{Z}$), the Heisenberg group of 3 by 3 upper diagonal matrices over the integers with 1 on diagonal. Take $G_{2}$ = $\mathbb{Z}^{3}$ , $K$ = $\mathbb{Z}$ and $L =\mathbb{Z}^{2}$. Certainly $G_{1}$ and $G_{2}$ are not quasi-isometric because $G_{1}$ and $G_{2}$ have different growth degree.