Do all right orderable groups have the Haagerup property?
Recall that a group is right orderable if there exists a total order $\leq$ on it such that $a\leq b\Rightarrow ac\leq bc$. This property is important for its connection to conjectures regarding group rings. Recall also that a group has the Haagerup property if it has a proper 1-cocycle. This property is e.g. relevant to Sorin Popa's deformation/rigidity techniques and to the Baum-Connes conjecture.