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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 Amenability of groups

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This reminds me the notion of pseudo-amenability, given by H. G. Dales (Proc. LMS 89, (2004)). Pseudo-amenability is a property such that (1) it's (formally) weaker than amenability; (2) it passes to a subgroup; (3) non-cyclic free groups do not have it; (4) not known whether it implies amenability. (Isn't there any (meta) technical term for such properties, say, satisfying 1-3 ?) If I remember correctly, Dales once gave a talk listing similar properties. –  Narutaka OZAWA Nov 9 '10 at 23:47
    
I have deleted two comments I made which were based on me misremembering what I had been told in conversation, and which I therefore want to remove to avoid confusing anyone who reads this later. (The first of these comments is what Andreas was responding to.) –  Yemon Choi Nov 12 '10 at 18:26

1 Answer 1

up vote 6 down vote accepted

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.

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This will even work for $C < \sqrt{2}$. One considers the closed convex hull $K$ of the orbit $\pi(G)\xi$. Then $G$ fixes the vector $\xi_0 \in K$ of minimal norm, and $\xi_0 \not= 0$ since one computes directly $2 \Re(\langle \eta, \xi \rangle) \geq (2 - C^2) > 0$ for all $\eta \in K$. –  Jesse Peterson Feb 10 '11 at 23:21

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