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 $T$-Følner 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 ${\lim }_{n\to \infty }{\sup }_{g\in K}\frac{\mu (T_g{A}_n△A_n)}{\mu (A_n)}=0$ 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 $S$-Følner sequence and such that $S\upharpoonright H$ is weakly mixing. These refine some recent results by Delabie-Jolissaint-Zumbrunnen and Jolissaint.

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