Flexible Hilbert-Schmidt stability versus hyperlinearity for property (T) groups

Abstract

We prove a statement concerning hyperlinearity for central extensions of property (T) groups in the presence of flexible HS-stability, and more generally, weak ucp-stability. Notably, this result is applied to show that if $\text{Sp}_{2g} (\mathbb Z)$ is flexibly HS-stable, then there exists a non-hyperlinear group. Further, the same phenomenon is shown to hold generically for random groups sampled in Gromov's density model, as well as all infinitely presented property (T) groups. This gives new directions for the possible existence of a non-hyperlinear group. Our results yield Hilbert-Schmidt analogues for Bowen and Burton's work relating flexible P-stability of $\text{PSL}_n(\mathbb Z)$ and the existence of non-sofic groups.