Żuk proved that if a finitely generated group admits a Cayley graph such that the Laplacian on the links of this Cayley graph has a spectral gap $>\frac{1}{2}$, then the group has property (T), or equivalently, every affine isometric action of the group on a Hilbert space has a fixed point. We prove that the same holds for affine isometric actions of the group on a uniformly curved Banach space (for example an $L^p$-space with $1<p<\infty$ or an interpolation space between a Hilbert space and an arbitrary Banach space) as soon as the Laplacian on the links has a two-sided spectral gap $>1-𝜀$.

This criterion applies to random groups in the triangular density model for densities $>\frac{1}{3}$. In this way, we are able to generalize recent results of Druţu and Mackay to affine isometric actions of random groups on uniformly curved Banach spaces. Also, in the setting of actions on $L^p$-spaces, our results are quantitatively stronger, even in the case $p=2$. This naturally leads to new estimates on the conformal dimension of the boundary of random groups in the triangular model.

Additionally, we obtain results on the eigenvalues of the $p$-Laplacian on graphs, and on the spectrum and degree distribution of Erdős–Rényi graphs.

Żuk证明，如果有限生成的群接纳Cayley图，使得该Cayley图的链接上的拉普拉斯算子具有谱隙 $>\frac{\mathrm{1个}}{2}$，则该组具有属性（T），或者等效地，该组在希尔伯特空间上的每个仿射等距作用都具有一个固定点。我们证明了在均匀弯曲的Banach空间上该组的仿射等距作用同样成立（例如${\mathrm{大号}}^p$-空间与 $\mathrm{1个}<p<\infty$ 或链接上的拉普拉斯算子具有两侧频谱间隙后立即在Hilbert空间和任意Banach空间之间进行插值空间） $>\mathrm{1个}-𝜀$。

此标准适用于三角形密度模型中的随机组 $>\frac{\mathrm{1个}}{3}$。通过这种方式，我们能够将Druţu和Mackay的最新结果推广到在均匀弯曲的Banach空间上仿射随机组的等距作用。另外，在${\mathrm{大号}}^p$-空间，即使在这种情况下，我们的结果在数量上也更强大 $p=2$。这自然会导致对三角模型中随机组的边界的保形维数进行新的估计。

此外，我们获得了关于特征值的结果。 $p$图上的拉普拉斯算子，以及Erdős–Rényi图的频谱和度数分布。