Near coverings and cosystolic expansion

Abstract

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 r X 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( √ 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)).

Stability of Homomorphisms, Coverings and Cocycles I: Equivalence

“…This idea of encoding graph coverings using 1-cochains is classical (cf. [Hat02, AL02, ALMR01]), and was specifically used in Dinur-Meshulam [DM22] in a similar way.…”

“…Covering stability, as in Definition 4.14, arises naturally in our proof strategy. In [DM22], Dinur-Meshulam studied a similar problem. The main difference between the Dinur-Meshulam setup and ours is that they think of the degree parameter n as a constant.…”

“…In this paper, we give an alternative setup for testing coverings, and consequently a different cohomological viewpoint from the one used in [DM22]. 1 In our setup, the parameter acquired by the cohomological viewpoint is equivalent to the covering testing parameter (as oppose to [DM22] in which they only bound one another). Moreover, both the cohomological and covering parameters are related to the homomorphism testing parameter of the fundamental group of the underlying complex.…”

“…Even more recently, Dinur-Meshulam [DM22] initiated the study of topological property testing. In their paper, they study whether being a topological covering is a testable property.…”

“…In this paper, we give an alternative setup for testing coverings, and consequently a different cohomological viewpoint from the one used in [DM22]. 1 In our setup, the parameter acquired by the cohomological viewpoint is equivalent to the covering testing parameter (as oppose to [DM22] in which they only bound one another).…”