Open Questions and Partial Results

Abstract

We have seen that, if a locally compact group G contains a closed noncom-pact subgroup H such that the pair (G, H) has relative property (T), then G may not have the Haagerup property. Hence we may ask whether relative property (T) (with respect to a noncompact subgroup) is the only obstruction to the Haagerup property. It was proved in Theorem 4.0.1 that the answer is positive for connected Lie groups¹.

Article Footnote

¹S.Popa suggested that, at the philosophical level, our question is analogous to von Neumann’s question: “Is the presence of a free nonabelian subgroup the only obstruction to amenability?”... and that both questions might share the same odd fate!