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 $exp(-O(r))$.

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 $exp(-O(r))$.