Approximation properties of simple Lie groups made discrete (Q2791845)

The following result is the main theorem of the paper under review.NEWLINENEWLINENEWLINE{Theorem.} Let \(G\) be a connected simple Lie group and let \(G_{\text d}\) be the group \(G\) equipped with the discrete topology. The following are equivalent: {\parindent=6mm \begin{itemize}\item[(1)] \(G\) is locally isomorphic to \(\mathrm{SO}(3)\), \(\mathrm{SL}(2,\mathbb R)\) or \(\mathrm{SL}(2,\mathbb C)\); \item[(2)] \(G_{\text d}\) has Haagerup property; \item[(3)] \(G_{\text d}\) is weakly amenable with constant \(1\); \item[(4)] \(G_{\text d}\) is weakly amenable; \item[(5)] \(G_{\text d}\) has weak Haagerup property with constant 1; \item[(6)] \(G_{\text d}\) has weak Haagerup property.NEWLINENEWLINE\end{itemize}}