TITLE: An example of a non-amenable exact group without  free subgroups.
QUESTION [20 upvotes]: A countable discrete group $\Gamma$ is said to be exact if it admits an amenable action on some compact space. 
So clearly amenable groups are exact, but large familes of non-amenable groups are as well.
For many of the families that I know of (ex. linear groups, hyperbolic groups) that are exact, they also satisfy the von Neumann conjecture (i.e. that if they are non-amenable then they have subgroup isomorphic to a free group.)
So my questions is:
Are there examples of exact groups that are non-amenable and do not contain free subgroups?

REPLY [3 votes]: It has been some years since this question was posted, but maybe (if you haven't seen it yet) you'll enjoy reading the new geometric solution for the von Neumann problem.
http://www.math.cornell.edu/~justin/Ftp/vN_fp.pdf