# Uniform Kazhdan Constants and Paradoxes of the Affine Plane

## TL;DR

This paper establishes uniform non-amenability and spectral gap properties for the affine group acting on the plane, using a novel uniform ping-pong argument for affine transformations.

## Contribution

It introduces a uniform quantitative ping-pong method for affine transformations, leading to new results on Kazhdan constants and non-amenability.

## Key findings

- Action of G on R^2 is uniformly non-amenable
- Quasi-regular representation of G has a uniform spectral gap
- Established a uniform ping-pong technique for affine transformations

## 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.