Haagerup property and Kazhdan pairs via ergodic infinite measure preserving actions

Abstract

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.