Denote $K, H$ to be countable discrete groups, then I am interested whether the crossed product group $G=H\rtimes_{\alpha} K$ is inner amenable or not.
For example, when $\alpha$ is trivial, $G=H\times K$, and if $H$ and $K$ is inner amenable, then $G$ is inner amenable by checking the definition of inner amenable. But in general, $G$ might be not inner amenable, for example, $G=M_4(\mathbb{Z})\rtimes SL_4(\mathbb{Z})$, which is a I.C.C.(infinite conjugacy class) group with Property (T) according to this paper.
My questions are the following.
Question 1:
Is it true that $G$ is an inner amenable group for any action $\alpha$ provided one of the following condition?
(c1) $H$ is inner amenable, $K$ is amenable.
(c2) $H$ is I.C.C. and inner amenable, $K$ is I.C.C. and inner amenable.
(c3) $H$ is I.C.C. and inner amenable, $K$ is amenable.
(c4) $H$ is I.C.C. and amenable, $K$ is I.C.C. and inner amenable.
Maybe another open-ended question is :
Question 2:
Find conditions, say (P), on the action $\alpha$ such that $(P)+(c4)\Longrightarrow G$ is inner amenable.
Any references, commentes are appreciated!
Remark:
1, This paper might be helpful, but I have not seen how to apply results in it to this question.
2, This question is motivated by a question in von Neumann algebras.
When $N$ is a II$_1$ factor with property Gamma, $K$ is an inner amenable group, suppose the crossed product $M=N\rtimes_{\alpha}K$ is still a II$_1$ factor, then when $M$ has property Gamma?
If $K$ is amenable, it is well-known $M$ always has property Gamma. Based on the close relation between $H$ is inner amenable and $L(H)$ has property Gamma, we might expect question 1 has a positive answer when (c3) holds.
(c4)(especially (c2)) does not imply $G$ is inner amenable in general, since for a non-amenable group $K$, the wreath product $H\wr K$ is never inner amenable by essentially the same argument following Proposition 2.23, part(b). And $\oplus H$ is I.C.C. iff $H$ is I.C.C.