  It is shown that a locally compact second countable group $G$ has the
Haagerup property if and only if there exists a sharply weak mixing 0-type
measure preserving free $G$-action $T=(T_g)_{g\in G}$ on an infinite
$\sigma$-finite standard measure space $(X,\mu)$ admitting an exhausting
$T$-F{\o}lner sequence (i.e. a sequence $(A_n)_{n=1}^\infty$ of measured
subsets of finite measure such that $A_1\subset A_2\subset\cdots$,
$\bigcup_{n=1}^\infty A_n=X$ and $\lim_{n\to\infty}\sup_{g\in
K}\frac{\mu(T_gA_n\triangle A_n)}{\mu(A_n)}= 0$ for each compact $K\subset G$).
It is also shown that a pair of groups $H\subset G$ has property (T) if and
only if there is a $\mu$-preserving $G$-action $S$ on $X$ admitting an
$S$-F{\o}lner sequence and such that $S\restriction H$ is weakly mixing. These
refine some recent results by Delabie-Jolissaint-Zumbrunnen and Jolissaint.
