High-dimensional expansion and soficity of groups

For d ≥ 4 and p a sufficiently large prime, we construct a lattice Γ ≤ PSp2d( Qₚ), such that its universal central extension cannot be sofic if Γ 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.