  Let $G=\mathrm{SL}(2,\mathbb{Z})\ltimes\mathbb{Z}^2$ and
$H=\mathrm{SL}(2,\mathbb{Z})$. We prove that the action
$G\curvearrowright\mathbb{R}^2$ is uniformly non-amenable and that the
quasi-regular representation of $G$ on $\ell^2(G/H)$ has a uniform spectral
gap. Both results are a consequence of a uniform quantitative form of ping-pong
for affine transformations, which we establish here.
