Agreement theorems for high dimensional expanders in the small soundness regime: the role of covers

Given a family $X$ of subsets of $[n]$ and an ensemble of local functions $\{f_s:s\to\Sigma\; | \; s\in X\}$, an agreement test is a randomized property tester that is supposed to test whether there is some global function $G:[n]\to\Sigma$ such that $f_s=G|_s$ for many sets $s$. A"classical"small-soundness agreement theorem is a list-decoding $(LD)$ statement, saying that \[\tag{$LD$} Agree(\{f_s\})>\varepsilon \quad \Longrightarrow \quad \exists G^1,\dots, G^\ell,\quad P_s[f_s\overset{0.99}{\approx}G^i|_s]\geq poly(\varepsilon),\;i=1,\dots,\ell. \] Such a statement is motivated by PCP questions and has been shown in the case where $X=\binom{[n]}k$, or where $X$ is a collection of low dimensional subspaces of a vector space. In this work we study small the case of on high dimensional expanders $X$. It has been an open challenge to analyze their small soundness behavior. Surprisingly, the small soundness behavior turns out to be governed by the topological covers of $X$.We show that: 1. If $X$ has no connected covers, then $(LD)$ holds, provided that $X$ satisfies an additional expansion property. 2. If $X$ has a connected cover, then $(LD)$ necessarily fails. 3. If $X$ has a connected cover (and assuming the additional expansion property), we replace the $(LD)$ by a weaker statement we call lift-decoding: \[ \tag{$LFD$} Agree(\{f_s\})>\varepsilon \Longrightarrow \quad \exists\text{ cover }\rho:Y\twoheadrightarrow X,\text{ and }G:Y(0)\to\Sigma,\text{ such that }\] \[P_{{\tilde s\twoheadrightarrow s}}[f_s \overset{0.99}{\approx} G|_{\tilde s}] \geq poly(\varepsilon),\] where ${\tilde s\twoheadrightarrow s}$ means that $\rho(\tilde s)=s$. The additional expansion property is cosystolic expansion of a complex derived from $X$ holds for the spherical building and for quotients of the Bruhat-Tits building.