## Haagerup property and Kazhdan pairs via ergodic infinite measure preserving actions

### Volume 265 / 2022

#### 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 $\lim _{n\to \infty }\sup _{g\in K}\frac {\mu (T_gA_n\bigtriangleup A_n)}{\mu (A_n)} = 0$ 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 $S {\restriction } H$ is weakly mixing. These refine some recent results by Delabie–Jolissaint–Zumbrunnen and Jolissaint.