Uniform Kazhdan Constants and Paradoxes of the Affine Plane

Lam Pham

Let $G=\mathrm{SL}(2,\mathbb{Z})⋉{\mathbb{Z}}^2$ and $H=\mathrm{SL}(2,\mathbb{Z})$. We prove that the action $G\curvearrowright {\mathbb{R}}^2$ is uniformly non-amenable and that…