High-dimensional expansion and soficity of groups

For $d \geq 4$ and $p$ a sufficiently large prime, we construct a lattice $\Gamma \leq {\rm PSp}_{2d}(\mathbb Q_p),$ such that its universal central extension cannot be sofic if $\Gamma$ 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.