标题： Ramanujan复形的准等距嵌入 显示英文标题

标题： Quasi-isometric embeddings of Ramanujan complexes

摘要： 拉马努金复形被定义为最优扩展器的高维类比，即拉马努金图。 它们被构造为对任何素数$p$的$\mathrm{PGL}_d(\mathbb{F}_p((y)))$的欧几里得建筑（也称为仿射建筑和布雷厄特-蒂茨建筑）的商空间，由卢博茨基-萨穆尔斯-维什内构造。 我们在大尺度几何上区分拉马努金复形。 更准确地说，我们证明如果$p$和$q$是不同的素数，那么相关的拉马努金复形不能彼此拟等距嵌入。 主要工具是 Khukhro-Valette 的方箱空间刚性和 Kleiner-Leeb 以及 Fisher-Whyte 的欧几里得建筑刚性。 显示英文摘要

摘要： Ramanujan complexes were defined as high dimensional analogues of the optimal expanders, Ramanujan graphs. They were constructed as quotients of the Euclidean building (also called the affine building and the Bruhat-Tits building) of $\mathrm{PGL}_d(\mathbb{F}_p((y)))$ for any prime $p$ by Lubotzky-Samuels-Vishne. We distinguish the Ramanujan complexes up to large-scale geometry. More precisely, we show that if $p$ and $q$ are distinct primes, then the associated Ramanujan complexes do not quasi-isometrically embed into one another. The main tools are the box space rigidity of Khukhro-Valette and the Euclidean building rigidity of Kleiner-Leeb and Fisher-Whyte.