High-dimensional expansion and soficity of groups

For $d\ge 4$ and $p$ a sufficiently large prime, we construct a lattice $\mathrm{\Gamma }\le {\mathrm{P}\mathrm{S}\mathrm{p}}_{2d}({\mathbb{Q}}_p),$ such that its universal central extension cannot be sofic if $\mathrm{\Gamma }$ satisfies some weak form of stability in permutations. In the proof, we make use of high-dimensional expansion phenomena and, extending results of Lubotzky, we construct new examples of cosystolic expanders over arbitrary finite abelian groups.