Let $G$ be non-amenable finitely generated group.


1) Is it true that there exists a sequence $S(n)$ of sets which generate $G$ and such that 


$\frac{1}{|S(n)|}||\sum\_{g\in S(n)} \lambda(g)||\rightarrow 0$ when $n\rightarrow \infty$.


2) The same as (1), but $S(n)$ is finite subset of $G$.


Here $\lambda:G\rightarrow B(l^2(G))$ is left regular representation of $G$.


Also 1) is reformulation of 2).


**Edit:** [here](https://mathoverflow.net/questions/55075/amenability-of-groups-iii) are some discussions on the question.



Amenability of groups
It was proved by Nagnibeda-Smirnova and myself (<http://arxiv.org/abs/1206.2183>) that if a group $\Gamma$ contains an infinite normal subgroup $N$ such that $\Gamma/N$ is not amenable then the question above for $\Gamma$ is true. This gave many examples of groups which are non-amenable and does not contain $\mathbb{F}\_2$ as a subgroup, such as Burnside and Golod-Shafarevych groups. As an easy consequence: if $\Gamma$ is not amenable then for $\Gamma\times \mathbb{Z}$ the question is true.


The question in the complete generality was finally solved by Andreas Thom (<http://arxiv.org/abs/1306.1767>)


There are applications to percolation theory (due to Nagnibeda-Pak) and to operator space analog of von Neumann's conjecture (due to Pisier) listed in the citations above. 


To my current knowledge (please correct me here), the question itself should be contributed either to Gilles Pisier or to Nagnibeda-Pak.


