Let G = SL(2, ℤ) ⋉ ℤ² and H = SL(2, ℤ). We prove that the action G ↷ ℝ² is uniformly non-amenable and that the quasi-regular representation of G on ℓ²(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.

中文翻译：

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仿射平面的一致Kazhdan常数和悖论

令G = SL（2 ，）²，H = SL（2 ，）。我们证明了动作ģ  ↷ℝ ²是均匀的非顺从和的准正则表示ģ上ℓ ²（G / H）具有均匀的频谱间隙。这两个结果都是乒乓球仿射变换的统一定量形式的结果，我们在这里建立了这种形式。