In the following discussion, we always assume $G$ is a countable discrete group.
I am learning about the group von Neumann algebra recently, We know that $L(G)$ is a factor if and only if $G$ is an i.c.c group. i.e. The conjugate class of every group element but the identity is infinite. In my opinion, it should mean that the group is far from Abelian.
Now, we may assume $G$ is an i.c.c group in the following discussion.
When $G$ is an i.c.c group, We know that $G$ is a group with Property (T) if and only if $L(G)$ is a von Neumann algebra with Property (T). $G$ is amenable if and only if $L(G)$ is amenable as a von Neumann algebra.
So It is natural to ask, Can we find some examples of i.c.c group with Property (T)? Can we find some examples of i.c.c group with amenability?
We know some example of i.c.c group such as $F_n$, the free group with $n$ generator, and $SL(n,\mathbb{Z})$ for $n$ odd. We can also prove that $SL(n,\mathbb{Z})$ for $n\geq 3$ is an example of group with Property (T), but $F_n$ doesn't have Property (T) since $\mathbb{Z}^n$ is a quotient of $F_n$.
In addition to that, we also know that $F_n$ ($n\geq 2$) and $SL(n,\mathbb{Z})$ ($n\geq 3$) are both non-ameanable group.
In summary, we have an example of i.c.c group with Property (T), which is $SL(n,\mathbb{Z})$ for $n\geq 3$ and $n$ odd.
My question is, can we have an example of an i.c.c group which is also amenable? It seems hard since the i.c.c property is a property which means the group should be far from Abelian and amenable property is a generalization of Abelian.
I also hope to have some more examples about i.c.c groups with Property (T) instead of the one I give beofre.
Any help will be truly grateful!