Brandeis Dynamics and Number Theory Seminar 2019-2020

(1) Uniform Kazhdan constants and paradoxes of the plane (2018), Preprint, Link

Abstract

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.