Hyperlinearity, stability and asymptotic spectral gap of higher rank lattices

We prove that if the group $\mathrm{SL}_2(\mathbb Z[1/p])$ is flexibly Hilbert--Schmidt stable for some prime $p$, then it admits a non-hyperlinear finite central extension. Consequently, a positive answer to the following question would yield an explicit example of a non-hyperlinear group: If two representations of the modular group $\mathrm{SL}_2(\mathbb{Z})$ almost agree on an Iwahori subgroup $B$, must they be close to representations that agree on $B$? More generally, we investigate spectral gap properties for asymptotic representations of higher rank lattices and groups with property (T:FD). In this setting, we prove that character rigidity is equivalent to a weak form of stability.