\(\mathrm{SL}_4(\mathbf{Z})\) is not purely matricial field (Q6621737)

Let \(\Gamma\) be a discrete group and let \(\{\rho_{i} \}^{\infty}_{i=1}\) be a sequence of finite dimensional unitary representations of \(\Gamma\). Then the sequence strongly converges to the regular representation (in symbols \(\rho_{i} \xrightarrow{\mathrm{strong}} \lambda_{\Gamma}\)) if for any \(z \in \mathbb{C}[\Gamma]\), \(\lim_{i \rightarrow \infty} ||\rho_{i}(z)||=\lambda(z)\), where \(\lambda_{\Gamma}: \Gamma \rightarrow U(\ell^{2}(\Gamma))\) is the left regular representation. The group \(\Gamma\) is purely matricial field if there is a sequence \(\{\rho_{i} \}^{\infty}_{i=1}\) of finite dimensional unitary representations such that \(\rho_{i} \xrightarrow{\mathrm{strong}} \lambda_{\Gamma}\).\N\NThe main result in the paper under review is Theorem 1: Every finite dimensional unitary representation of \(\mathrm{SL}_{4}(\mathbb{Z})\) contains a non-zero \(\mathrm{SL}_{2}(\mathbb{Z})\)-invariant vector.\N\NAs a consequence of Theorem 1, the authors prove (Corollary 3) that \(\mathrm{SL}_{4}(\mathbb{Z})\) is not purely matricial field.