For higher-rank lattices, Hilbert-Schmidt stability implies non-hyperlinearity of certain central extensions, and character rigidity is equivalent to hyperfinite Hilbert-Schmidt stability.
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Hyperlinearity, stability and asymptotic spectral gap of higher rank lattices
For higher-rank lattices, Hilbert-Schmidt stability implies non-hyperlinearity of certain central extensions, and character rigidity is equivalent to hyperfinite Hilbert-Schmidt stability.