发表机构
Institute of Mathematics of the Polish Academy of Sciences（波兰科学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明平方模型中随机群在密度大于5/12时渐近几乎必然具有Kazhdan性质(T)，并推广到L^p和F_X性质，给出边界共形维数界，提出基于谱链接准则的新方法。
AI 中文摘要
平方模型 $\mathcal{Q}(n,d)$ 中的随机群由一个有 $n$ 个生成元的展示以及一个长度为 4 的均匀随机循环约化关系集（共 $(2n-1)^{4d}$ 个）给出。我们证明，若 $d>\frac{5}{12}$，则 $\mathcal{Q}(n,d)$ 中的随机群渐近几乎必然（a.a.s.）具有 Kazhdan 性质 (T)。这为当 $d\in\left(\frac{5}{12}, \frac{1}{2}\right)$ 时提供了无限字双曲 Kazhdan 群的新构造。此外，同样对于 $d>\frac{5}{12}$，我们证明平方模型中的随机群 a.a.s. 对递增的 $p$ 值范围具有性质 $(\textrm{F}L^p)$，并对任何固定的均匀曲率 Banach 空间 $X$ 具有性质 $(\textrm{F}_X)$。这些结果导致对平方模型中随机群边界的共形维数的界。在此过程中，我们设计了一种方法，将性质 (T) 和其他不动点性质的谱链接准则应用于由长度为 4 的关系所呈现的群。为了将其用于平方模型，我们遵循 Broder-Shamir 工作的精神，通过迹方法验证所需的谱间隙假设。
英文摘要
A random group in the square model $\mathcal{Q}(n,d)$ is given by a presentation with $n$ generators and a uniformly random set of $(2n-1)^{4d}$ cyclically reduced relators of length 4 over them. We prove that if $d>\frac{5}{12}$, then a random group in $\mathcal{Q}(n,d)$ has Kazhdan's property (T) asymptotically almost surely (a.a.s.). This provides a new construction of infinite word-hyperbolic Kazhdan groups when $d\in\left(\frac{5}{12}, \frac{1}{2}\right)$. Moreover, also for $d>\frac{5}{12}$, we show that a random group in the square model a.a.s. has property $(\textrm{F}L^p)$ for an increasing range of values of $p$, and property $(\textrm{F}_X)$ for any fixed uniformly curved Banach space $X$. These results lead to bounds on the conformal dimension of the boundary of a random group in the square model. In the process we devise a method of applying the spectral link criteria for property (T), and other fixed point properties, to the groups presented by relations of length 4. In order to use it for the square model, we verify the required spectral gap hypothesis by the trace method, in the spirit of the work of Broder-Shamir.
