I am looking for a certain masa in a $II\_1$ factor which is singular and has nontrivial Takesaki invariant.
For this I am looking for an example of an inclusion of groups $H\subset G$ such that:


* $G$ is a countable icc (infinite conjugacy class) group
* $H$ is abelian
* $\forall g\in G-H,\{ hgh^{-1} |h\in H \}$ is infinite
* $ |H\backslash G/H| \geq 3$
* there exists $g\in G-H$ and $h\_1\neq h\_2\in H$ with $h\_1 g=gh\_2$.


Does such an example exist?



Does such a subgroup exist?
[generalization of Agol's answer]


Take $H$ a group and let $K$ act on $H$ by isomorphisms (write the action as $\sigma$) and consider $G=H\rtimes\_\sigma K$. Then


* condition 2 is satisfied if $H$ is abelian
* condition 4 is satisfied if $K$ contains at least 3 elements
* condition 5 is satisfied if $K$ acts non-trivially
* condition 3 is satisfied if $\{h^{-1}\sigma\_k(h) : h\in H\}$ is infinite for all $k \in K$
* condition 1 is satisfied if $K$ acts with infinite orbits on $H$ and condition 3 is satisfied.


