Question

Does this property characterizes amenability or there are examples of non-amenable groups satisfying it?

Let $G$ be finitely generated group.

Property:

There exists $C<1$ such that for every $S\subset G$ - finite set, there exists $F \subset G$ - finite, such that

$|sF \Delta F|\leq C\cdot |F|$ for every $s\in S$

Currently I don't see if it is possible to rebuilt sets $F$ in order to construct Folner sequence for $G$.

The question is related to http://mathoverflow.net/questions/44843/amenability-of-groups

Answer 1

If I am not mistaken the group should be amenable by the following arguments.

As was explained to me by Jesse Peterson, if $\pi:G\rightarrow B(H)$ is a representation of $G$ and there exists unit vector $\xi$ such that for every $g\in G$: $\|\pi(g)\xi-\xi\|\leq C$ then $\pi$ has an invariant vector.

Let $\xi_F=\frac{1}{\sqrt{F}}1_{F}$, then from the property we have $||\lambda(s)\xi_F-\xi_{F}||\leq C$ for every $s\in S$. Then taking $S_i$ as an increasing sequence such that $\cup S_i=G$ and ultralimit of $\lambda$, denote $\lambda_{\omega}:G\rightarrow B(l_2(G)^{\omega})$, we have the existence of a unit vector $\xi$ such that $||\lambda_{\omega}(g)\xi-\xi||\leq C$ for every $g\in G$. Thus $\lambda_{\omega}$ has an invariant vector, which implies that $\lambda$ has a sequence of almost invariant vectors. The last one is equivalent to amenability.