I have to say that this article http://arxiv.org/abs/1506.00887 has at least one mistake.
Lemma 3.5, Counterexample - take group $G=$ sequences of integer numbers $=\mathbb{Z}^\mathbb{N}$.
$\ell$ and $m$ are right and left shift respectively.
Then $m\circ\ell=id$, so $Image(e)=G$.
