oa.operator algebras - Is there an i.c.c. nonamenable simple group that is inner amenable?

Hey Jon

So my initial thought would be no.

First, in full generality every group is virtually inner-amenable. Meaning that for any group $G$, the group $G times mathbb{Z}/2mathbb{Z}$ is inner amenable. In fact, any non-icc group is inner amenable just by taking the mean to be the counting measure on a finite conjugacy class, and 0 elsewhere.

Even if we restrict to icc groups then, for any icc group $G$, $Gtimes S_infty$ (or just choose the second group to be anything inner amenable) is still inner amenable.

And because the group is formed as a direct product there is not any way for the generators of $S_infty$ to sort of "slow down" the growth in the $G$ factor.

Now a final way to maybe make something out of this is to ask

"If $G$ is inner amenable and along with all of its quotients, then is there a growth contsraint."

This will get rid of the examples above. Amenable groups fall into this class, and I would be willing to bet that there are others as well (if anyone knows examples that would be nice) but I can't think of any on the spot.

AS for this class.... I have no idea.