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：我们给出了几类高维展开复形的余闭链展开常数的新界，以及齐次几何格序复形的已知上同调边界展开常数（包括$SL_n(F_q)$的球面建筑）的改进界。该改进适用于Lubotzky、Samuels和Vishne以及Kaufman和Oppenheim构造的高维展开复形。我们的新展开常数不依赖于复形的度数、维数，也不依赖于系数群。这改进了Gromov拓扑重叠常数以及Dinur和Meshulam覆盖稳定性的界，可能对一致性检验有应用价值。相比之下，现有界随环境维数呈指数衰减（对球面建筑），且随度数线性衰减（对所有已知有界度高维展开复形）。我们的结果基于多项新技术：* 我们发展了一种新的“颜色限制”技术，通过将多部复形限制在其颜色类的小随机子集上，证明无维数展开性质。* 我们给出了Evra和Kaufman局部到全局定理的新“谱”证明，得到更优的界并消除了对度数的依赖。该定理利用链的上同调边界展开和谱展开来约束复形的余闭链展开。* 我们通过构造新颖的极短锥族，推导出球面建筑（以及任何齐次几何格序复形）的上同调边界展开的绝对界。