A closer look at the non-Hopfianness of $BS(2,3)$

The Baumslag-Solitar group $BS(2,3)$, is a so-called non-Hopfian group, meaning that it has an epimorphism $\varphi$ onto itself, that is not injective. In particular this is equivalent to saying that $BS(2,3)$ has a non-trivial quotient that is isomorphic to itself. As a consequence the Cayley graph of $BS(2,3)$ has a quotient that is isomorphic to itself up to change of generators. We describe this quotient on the graph-level and take a closer look at the most common epimorphism $\varphi$. We show its kernel is a free group of infinite rank with an explicit set of generators. Finally we show how $\varphi$ appears as a morphism on fundamental groups induced by some continuous map. This point of view was communicated to the author by Gilbert Levitt.