标题： 新的余同调高维膨胀器来自KMS群 显示英文标题

标题： New cosystolic high-dimensional expanders from KMS groups

摘要： 余同调扩张是单纯复形的Cheeger常数的一个高维推广。 最初，这一概念的动机在于它暗示了拓扑重叠性质的存在，但最近的研究表明，它与理论计算机科学中的问题有关，例如低误差率下的列表一致性扩张和一致性扩张。 目前仅有少数几种高维余同调扩张的构造方法，并且在维度大于$2$的情况下，在我们这项工作之前已知的唯一构造方法是仿射建筑商的（余维数为1）骨架。 在本文中，我们首次提出了任意维度下余同调扩张的陪集复形构造方法。 我们的构造方法比依赖于仿射建筑商商群的先前方法更加对称，也更基本。 我们所考虑的陪集复形来源于Kac--Moody--Steinberg（KMS）群的有限商群，并被称为KMS复形。 KMS复形是由Grave de Peralta和Valentiner-Branth最近的工作引入的，他们证明了这些复形是局部谱扩张。 我们的研究结果表明，满足某些次要条件的KMS复形可以产生任意维度和任意有限生成交换系数群的有界度余同调扩张的无穷族。 通过观察到KMS复形的适当链接是球形建筑中对立复形的连接，我们得到了这一结果。 为了证明这些对立复形是余边界扩张器，我们开发了一种新的方法，通过迭代添加顶点集合来构造锥函数。 因此，我们证明了KMS复形的链接是余边界扩张器。 利用先前的局部-全局结果，我们获得了KMS复形的（余维数为1）骨架的余同调扩张。 显示英文摘要

摘要： Cosystolic expansion is a high-dimensional generalization of the Cheeger constant for simplicial complexes. Originally, this notion was motivated by the fact that it implies the topological overlapping property, but more recently it was shown to be connected to problems in theoretical computer science such as list agreement expansion and agreement expansion in the low soundness regime. There are only a few constructions of high-dimensional cosystolic expanders and, in dimension larger than $2$, the only known constructions prior to our work were (co-dimension 1)-skeletons of quotients of affine buildings. In this paper, we give the first coset complex construction of cosystolic expanders for an arbitrary dimension. Our construction is more symmetric and arguably more elementary than the previous constructions relying on quotients of affine buildings. The coset complexes we consider arise from finite quotients of Kac--Moody--Steinberg (KMS) groups and are known as KMS complexes. KMS complexes were introduced in recent work by Grave de Peralta and Valentiner-Branth where it was shown that they are local-spectral expanders. Our result is that KMS complexes, satisfying some minor condition, give rise to infinite families of bounded degree cosystolic expanders of arbitrary dimension and for any finitely generated Abelian coefficient group. This result is achieved by observing that proper links of KMS complexes are joins of opposition complexes in spherical buildings. In order to show that these opposition complexes are coboundary expanders, we develop a new method for constructing cone functions by iteratively adding sets of vertices. Hence we show that the links of KMS complexes are coboundary expanders. Using the prior local-to-global results, we obtain cosystolic expansion for the (co-dimension 1)-skeletons of the KMS complexes.