We give new bounds on the cosystolic expansion constants of several families of high dimensional expanders, and the known coboundary expansion constants of order complexes of homogeneous geometric lattices, including the spherical building of $SL_n(F_q)$. The improvement applies to the high dimensional expanders constructed by Lubotzky, Samuels and Vishne, and by Kaufman and Oppenheim. Our new expansion constants do not depend on the degree of the complex nor on its dimension, nor on the group of coefficients. This implies improved bounds on Gromov's topological overlap constant, and on Dinur and Meshulam's cover stability, which may have applications for agreement testing. In comparison, existing bounds decay exponentially with the ambient dimension (for spherical buildings) and in addition decay linearly with the degree (for all known bounded-degree high dimensional expanders). Our results are based on several new techniques: * We develop a new "color-restriction" technique which enables proving dimension-free expansion by restricting a multi-partite complex to small random subsets of its color classes. * We give a new "spectral" proof for Evra and Kaufman's local-to-global theorem, deriving better bounds and getting rid of the dependence on the degree. This theorem bounds the cosystolic expansion of a complex using coboundary expansion and spectral expansion of the links. * We derive absolute bounds on the coboundary expansion of the spherical building (and any order complex of a homogeneous geometric lattice) by constructing a novel family of very short cones.

翻译：我们给出了若干族高维扩张图的新余循环扩张常数上界，以及包括$SL_n(F_q)$球面建筑物在内的齐次几何格序复形的已知上同调边界扩张常数的新上界。该改进适用于Lubotzky、Samuels与Vishne以及Kaufman与Oppenheim构造的高维扩张图。我们的新扩张常数既不依赖于复形的度数，也不依赖于其维数或系数群。这改进了Gromov拓扑重叠常数以及Dinur与Meshulam覆盖稳定性的上界，可能在一致性检验中具有应用潜力。相比之下，现有界值随环境维数呈指数衰减（对球面建筑物而言），且对已知有界度数高维扩张图还随度数线性衰减。我们的结果基于若干新技术：* 我们发展了一种新的"颜色限制"技术，通过将多部图复形限制到其颜色类的小随机子集上来证明与维数无关的扩张性。* 我们为Evra与Kaufman的局部到全局定理给出了新的"谱"证明，得到了更优上界并消除了对度数的依赖。该定理利用链接的上同调边界扩张与谱扩张来约束复形的余循环扩张。* 我们通过构造一类全新的极短锥，推导了球面建筑物（及任意齐次几何格序复形）的上同调边界扩张的绝对上界。