Cocycle expansion for quotients of Bruhat–Tits buildings with permutation coefficients

Let $G$ be a simple $p$-adic Lie group of rank $d\geq 3$, let $p\geq p(d)\gg0$, let $\Gamma$ be a lattice in $G$, and let $\mathcal{B}$ be the Bruhat–Tits building associated with $G$. The notation $h_1({}_\Gamma\setminus^\mathcal{B},{\rm Sym})$ denotes the relevant first cocycle Cheeger constant with permutation coefficients.

Cocycle-expansion conjecture. For every such lattice $\Gamma$,

$$h_1({}_\Gamma\setminus^\mathcal{B},{\rm Sym})>0.$$

This would establish cocycle expansion with permutation coefficients for quotients of Bruhat–Tits buildings of high-rank simple $p$-adic Lie groups, a proposed next step beyond the paper’s analysis of large cosystoles and property $(\tau)$.