Uniform Kazhdan Constants and Paradoxes of the Affine Plane

Let $G=\mathrm{S}\mathrm{L}(2,\mathbb{Z})⋉{\mathbb{Z}}^2$ and $H=\mathrm{S}\mathrm{L}(2,\mathbb{Z})$. We prove that the action $G\curvearrowright {\mathbb{R}}^2$ is uniformly non-amenable and that the quasi-regular representation of $G$ on ${\mathrm{\ell }}^2(G\mathrm{/}H)$ 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.