Haagerup property and Kazhdan pairs via ergodic infinite measure preserving actions

TL;DR

This paper characterizes the Haagerup property and Kazhdan pairs for groups via ergodic measure-preserving actions with specific mixing and Følner sequence properties, refining recent results in the field.

Contribution

It provides new characterizations of the Haagerup property and property (T) using ergodic actions with Følner sequences, connecting group properties to dynamical systems.

Findings

Haagerup property characterized by sharply weak mixing actions with Følner sequences.

Kazhdan pairs characterized by actions with Følner sequences where restrictions are weakly mixing.

Refines recent results by Delabie-Jolissaint-Zumbrunnen and Jolissaint.

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.