  The weak Haagerup property for locally compact groups and the weak Haagerup
constant was recently introduced by the second author. The weak Haagerup
property is weaker than both weak amenability introduced by Cowling and the
first author and the Haagerup property introduced by Connes and Choda.
  In this paper it is shown that a connected simple Lie group G has the weak
Haagerup property if and only if the real rank of G is zero or one. Hence for
connected simple Lie groups the weak Haagerup property coincides with weak
amenability. Moreover, it turns out that for connected simple Lie groups the
weak Haagerup constant coincides with the weak amenability constant, although
this is not true for locally compact groups in general.
  It is also shown that the semidirect product of R^2 by SL(2,R) does not have
the weak Haagerup property.
