We introduce and study swap cosystolic expansion, a new expansion property of simplicial complexes. We prove lower bounds for swap coboundary expansion of spherical buildings and use them to lower bound swap cosystolic expansion of the LSV Ramanujan complexes. Our motivation is the recent work (in a companion paper) showing that swap cosystolic expansion implies agreement theorems. Together the two works show that these complexes support agreement tests in the low acceptance regime. Swap cosystolic expansion is defined by considering, for a given complex $X$, its faces complex $F^r X$, whose vertices are $r$-faces of $X$ and where two vertices are connected if their disjoint union is also a face in $X$. The faces complex $F^r X$ is a derandomizetion of the product of $X$ with itself $r$ times. The graph underlying $F^rX$ is the swap walk of $X$, known to have excellent spectral expansion. The swap cosystolic expansion of $X$ is defined to be the cosystolic expansion of $F^r X$. Our main result is a $\exp(-O(\sqrt r))$ lower bound on the swap coboundary expansion of the spherical building and the swap cosystolic expansion of the LSV complexes. For more general coboundary expanders we show a weaker lower bound of $exp(-O(r))$.

翻译：我们引入并研究了交换余循环扩张（swap cosystolic expansion），这是一种新的单纯复形扩张性质。我们证明了球面建筑的交换余边界扩张的下界，并利用这些下界得到LSV拉马努金复形的交换余循环扩张的下界。我们的动机源于近期工作（在姊妹论文中）表明：交换余循环扩张可推出一致性定理。这两项工作共同表明，这些复形在低接受率体制下支持一致性测试。交换余循环扩张的定义基于：对给定复形$X$，考虑其面复形$F^r X$，其顶点为$X$的$r$维面，当两个顶点的无交并仍是$X$中的面时，它们相连。面复形$F^r X$是$X$与自身$r$次乘积的去随机化版本。$F^rX$的底层图是$X$的交换游走，已知其具有优异的光谱扩张性质。$X$的交换余循环扩张定义为$F^r X$的余循环扩张。我们的主要结果是：球面建筑的交换余边界扩张和LSV复形的交换余循环扩张存在$\exp(-O(\sqrt r))$下界。对于更一般的余边界扩张器，我们证明了较弱的下界$\exp(-O(r))$。