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

TL;DR

This paper explores the relationship between flexible HS-stability and hyperlinearity in property (T) groups, showing that certain stability assumptions imply the existence of non-hyperlinear groups, with implications for the broader understanding of group stability.

Contribution

It establishes a connection between flexible HS-stability and the existence of non-hyperlinear groups, extending the theory to random groups and infinitely presented property (T) groups.

Findings

If Sp_{2g}(Z) is flexibly HS-stable, then a non-hyperlinear group exists.

The phenomenon applies to random groups in Gromov's density model.

Results provide Hilbert-Schmidt analogues for stability and soficity questions.

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.