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 $Image(e)=G$.